All the Pure Nash Equiilbria, and ONE Mixed One, for von Neumann Poker with \

    number of cards from 2 to, 10, and bet sizes from, 1,  to, 5



                              By Shalosh B. Ekhad



Player 1 and Player 2 each get (different) cards from a deck of n cards numb\
    ered 1,..., n. They each see their own card, but not the other's



                       They each put 1 dollar in the pot



Player 1 can either check, in which case the cards are compared, and whoever\
     has the larger card gets the pot (and hence wins 1 dollar), or bet an a\
    dditional amount of b dollars



Now Player 2 can decide to cut his losses, and fold, and the pot goes immedi\
    ately to player 1, who is going to make a dollar, and player 2 is going \
    to lose a dollar



                    or else to also put b dollars in the pot



Now the cards are compared, and whoever has the larger card gets the pot, wi\
    nning b+1 dollars (and the other player lost b+1 dollars



Here all the pure Nash Equilibria, and ONE Mixed one, for card sizes from 2 to,

    10, and bet sizes from, 1,  to, 5



                  --------------------------------------------

                  --------------------------------------------

           If the deck has, 2, distinct cards and the bet size is, 1



                         There are, 2, Nash equilibria

                                 here they are

                                  NE number, 1

Player 1 bets if his card is in the set, {},

    and checks otherwise, while Player 2 calls if his card is in the set, {2},

    and folds otherwies , the value of the game is, 0

                                  NE number, 2

Player 1 bets if his card is in the set, {2},

    and checks otherwise, while Player 2 calls if his card is in the set, {2},

    and folds otherwies , the value of the game is, 0

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 1,

    and fold with probability, 0

        The value of this pair of strategies is, 0, and in decmimals, 0.

                  --------------------------------------------

           If the deck has, 2, distinct cards and the bet size is, 2



                         There are, 2, Nash equilibria

                                 here they are

                                  NE number, 1

Player 1 bets if his card is in the set, {},

    and checks otherwise, while Player 2 calls if his card is in the set, {2},

    and folds otherwies , the value of the game is, 0

                                  NE number, 2

Player 1 bets if his card is in the set, {2},

    and checks otherwise, while Player 2 calls if his card is in the set, {2},

    and folds otherwies , the value of the game is, 0

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 1,

    and fold with probability, 0

        The value of this pair of strategies is, 0, and in decmimals, 0.

                  --------------------------------------------

           If the deck has, 2, distinct cards and the bet size is, 3



                         There are, 2, Nash equilibria

                                 here they are

                                  NE number, 1

Player 1 bets if his card is in the set, {},

    and checks otherwise, while Player 2 calls if his card is in the set, {2},

    and folds otherwies , the value of the game is, 0

                                  NE number, 2

Player 1 bets if his card is in the set, {2},

    and checks otherwise, while Player 2 calls if his card is in the set, {2},

    and folds otherwies , the value of the game is, 0

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 1,

    and fold with probability, 0

        The value of this pair of strategies is, 0, and in decmimals, 0.

                  --------------------------------------------

           If the deck has, 2, distinct cards and the bet size is, 4



                         There are, 2, Nash equilibria

                                 here they are

                                  NE number, 1

Player 1 bets if his card is in the set, {},

    and checks otherwise, while Player 2 calls if his card is in the set, {2},

    and folds otherwies , the value of the game is, 0

                                  NE number, 2

Player 1 bets if his card is in the set, {2},

    and checks otherwise, while Player 2 calls if his card is in the set, {2},

    and folds otherwies , the value of the game is, 0

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 1,

    and fold with probability, 0

        The value of this pair of strategies is, 0, and in decmimals, 0.

                  --------------------------------------------

           If the deck has, 2, distinct cards and the bet size is, 5



                         There are, 2, Nash equilibria

                                 here they are

                                  NE number, 1

Player 1 bets if his card is in the set, {},

    and checks otherwise, while Player 2 calls if his card is in the set, {2},

    and folds otherwies , the value of the game is, 0

                                  NE number, 2

Player 1 bets if his card is in the set, {2},

    and checks otherwise, while Player 2 calls if his card is in the set, {2},

    and folds otherwies , the value of the game is, 0

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 1,

    and fold with probability, 0

        The value of this pair of strategies is, 0, and in decmimals, 0.

                  --------------------------------------------

                  --------------------------------------------

           If the deck has, 3, distinct cards and the bet size is, 1



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 1/3,

    and check with probability, 2/3

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 1/3,

    and fold with probability, 2/3

If Player 2 got card, 3,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 1/18, and in decmimals, 0.05555555556

                  --------------------------------------------

           If the deck has, 3, distinct cards and the bet size is, 2



                         There are, 2, Nash equilibria

                                 here they are

                                  NE number, 1

Player 1 bets if his card is in the set, {},

    and checks otherwise, while Player 2 calls if his card is in the set, {3},

    and folds otherwies , the value of the game is, 0

                                  NE number, 2

Player 1 bets if his card is in the set, {3},

    and checks otherwise, while Player 2 calls if his card is in the set, {3},

    and folds otherwies , the value of the game is, 0

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 1,

    and fold with probability, 0

        The value of this pair of strategies is, 0, and in decmimals, 0.

                  --------------------------------------------

           If the deck has, 3, distinct cards and the bet size is, 3



                         There are, 2, Nash equilibria

                                 here they are

                                  NE number, 1

Player 1 bets if his card is in the set, {},

    and checks otherwise, while Player 2 calls if his card is in the set, {3},

    and folds otherwies , the value of the game is, 0

                                  NE number, 2

Player 1 bets if his card is in the set, {3},

    and checks otherwise, while Player 2 calls if his card is in the set, {3},

    and folds otherwies , the value of the game is, 0

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 1,

    and fold with probability, 0

        The value of this pair of strategies is, 0, and in decmimals, 0.

                  --------------------------------------------

           If the deck has, 3, distinct cards and the bet size is, 4



                         There are, 2, Nash equilibria

                                 here they are

                                  NE number, 1

Player 1 bets if his card is in the set, {},

    and checks otherwise, while Player 2 calls if his card is in the set, {3},

    and folds otherwies , the value of the game is, 0

                                  NE number, 2

Player 1 bets if his card is in the set, {3},

    and checks otherwise, while Player 2 calls if his card is in the set, {3},

    and folds otherwies , the value of the game is, 0

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 1,

    and fold with probability, 0

        The value of this pair of strategies is, 0, and in decmimals, 0.

                  --------------------------------------------

           If the deck has, 3, distinct cards and the bet size is, 5



                         There are, 2, Nash equilibria

                                 here they are

                                  NE number, 1

Player 1 bets if his card is in the set, {},

    and checks otherwise, while Player 2 calls if his card is in the set, {3},

    and folds otherwies , the value of the game is, 0

                                  NE number, 2

Player 1 bets if his card is in the set, {3},

    and checks otherwise, while Player 2 calls if his card is in the set, {3},

    and folds otherwies , the value of the game is, 0

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 1,

    and fold with probability, 0

        The value of this pair of strategies is, 0, and in decmimals, 0.

                  --------------------------------------------

                  --------------------------------------------

           If the deck has, 4, distinct cards and the bet size is, 1



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 1/3,

    and check with probability, 2/3

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 4,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 1/12, and in decmimals, 0.08333333333

                  --------------------------------------------

           If the deck has, 4, distinct cards and the bet size is, 2



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 1/2,

    and check with probability, 1/2

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 1/2,

    and fold with probability, 1/2

If Player 2 got card, 4,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 1/12, and in decmimals, 0.08333333333

                  --------------------------------------------

           If the deck has, 4, distinct cards and the bet size is, 3



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 3/5,

    and check with probability, 2/5

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 1/5,

    and fold with probability, 4/5

If Player 2 got card, 4,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 1/20, and in decmimals, 0.05000000000

                  --------------------------------------------

           If the deck has, 4, distinct cards and the bet size is, 4



                         There are, 2, Nash equilibria

                                 here they are

                                  NE number, 1

Player 1 bets if his card is in the set, {},

    and checks otherwise, while Player 2 calls if his card is in the set, {4},

    and folds otherwies , the value of the game is, 0

                                  NE number, 2

Player 1 bets if his card is in the set, {4},

    and checks otherwise, while Player 2 calls if his card is in the set, {4},

    and folds otherwies , the value of the game is, 0

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 1,

    and fold with probability, 0

        The value of this pair of strategies is, 0, and in decmimals, 0.

                  --------------------------------------------

           If the deck has, 4, distinct cards and the bet size is, 5



                         There are, 2, Nash equilibria

                                 here they are

                                  NE number, 1

Player 1 bets if his card is in the set, {},

    and checks otherwise, while Player 2 calls if his card is in the set, {4},

    and folds otherwies , the value of the game is, 0

                                  NE number, 2

Player 1 bets if his card is in the set, {4},

    and checks otherwise, while Player 2 calls if his card is in the set, {4},

    and folds otherwies , the value of the game is, 0

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 1,

    and fold with probability, 0

        The value of this pair of strategies is, 0, and in decmimals, 0.

                  --------------------------------------------

                  --------------------------------------------

           If the deck has, 5, distinct cards and the bet size is, 1



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 1/3,

    and check with probability, 2/3

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 2/3,

    and fold with probability, 1/3

If Player 2 got card, 4,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 5,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 1/12, and in decmimals, 0.08333333333

                  --------------------------------------------

           If the deck has, 5, distinct cards and the bet size is, 2



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 1/2,

    and check with probability, 1/2

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 5,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 1/10, and in decmimals, 0.1000000000

                  --------------------------------------------

           If the deck has, 5, distinct cards and the bet size is, 3



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 3/5,

    and check with probability, 2/5

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 3/5,

    and fold with probability, 2/5

If Player 2 got card, 5,  he should call With probability, 1,

    and fold with probability, 0

The value of this pair of strategies is, 9/100, and in decmimals, 0.09000000000

                  --------------------------------------------

           If the deck has, 5, distinct cards and the bet size is, 4



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 2/3,

    and check with probability, 1/3

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 1/3,

    and fold with probability, 2/3

If Player 2 got card, 5,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 1/15, and in decmimals, 0.06666666667

                  --------------------------------------------

           If the deck has, 5, distinct cards and the bet size is, 5



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 5/7,

    and check with probability, 2/7

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 1/7,

    and fold with probability, 6/7

If Player 2 got card, 5,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 1/28, and in decmimals, 0.03571428571

                  --------------------------------------------

                  --------------------------------------------

           If the deck has, 6, distinct cards and the bet size is, 1



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 2/3,

    and check with probability, 1/3

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 1,

    and check with probability, 0

If Player 1  got card, 6,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 1/3,

    and fold with probability, 2/3

If Player 2 got card, 4,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 5,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 6,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 4/45, and in decmimals, 0.08888888889

                  --------------------------------------------

           If the deck has, 6, distinct cards and the bet size is, 2



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 1/2,

    and check with probability, 1/2

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 1/2,

    and fold with probability, 1/2

If Player 2 got card, 5,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 6,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 1/10, and in decmimals, 0.1000000000

                  --------------------------------------------

           If the deck has, 6, distinct cards and the bet size is, 3



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 3/5,

    and check with probability, 2/5

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 5,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 6,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 1/10, and in decmimals, 0.1000000000

                  --------------------------------------------

           If the deck has, 6, distinct cards and the bet size is, 4



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 2/3,

    and check with probability, 1/3

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 5,  he should call With probability, 2/3,

    and fold with probability, 1/3

If Player 2 got card, 6,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 4/45, and in decmimals, 0.08888888889

                  --------------------------------------------

           If the deck has, 6, distinct cards and the bet size is, 5



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 5/7,

    and check with probability, 2/7

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 5,  he should call With probability, 3/7,

    and fold with probability, 4/7

If Player 2 got card, 6,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 1/14, and in decmimals, 0.07142857143

                  --------------------------------------------

                  --------------------------------------------

           If the deck has, 7, distinct cards and the bet size is, 1



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 2/3,

    and check with probability, 1/3

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 1,

    and check with probability, 0

If Player 1  got card, 7,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 5,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 6,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 7,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 2/21, and in decmimals, 0.09523809524

                  --------------------------------------------

           If the deck has, 7, distinct cards and the bet size is, 2



                         There are, 3, Nash equilibria

                                 here they are

                                  NE number, 1

Player 1 bets if his card is in the set, {1, 6, 7},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {3, 6, 7}, and folds otherwies , the value of the game is, 2/21

                                  NE number, 2

Player 1 bets if his card is in the set, {1, 6, 7},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {4, 6, 7}, and folds otherwies , the value of the game is, 2/21

                                  NE number, 3

Player 1 bets if his card is in the set, {1, 6, 7},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {5, 6, 7}, and folds otherwies , the value of the game is, 2/21

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 1/2,

    and check with probability, 1/2

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 7,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 5,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 6,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 7,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 2/21, and in decmimals, 0.09523809524

                  --------------------------------------------

           If the deck has, 7, distinct cards and the bet size is, 3



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 3/5,

    and check with probability, 2/5

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 7,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 5,  he should call With probability, 2/5,

    and fold with probability, 3/5

If Player 2 got card, 6,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 7,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 1/10, and in decmimals, 0.1000000000

                  --------------------------------------------

           If the deck has, 7, distinct cards and the bet size is, 4



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 2/3,

    and check with probability, 1/3

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 7,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 5,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 6,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 7,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 2/21, and in decmimals, 0.09523809524

                  --------------------------------------------

           If the deck has, 7, distinct cards and the bet size is, 5



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 5/7,

    and check with probability, 2/7

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 7,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 5,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 6,  he should call With probability, 5/7,

    and fold with probability, 2/7

If Player 2 got card, 7,  he should call With probability, 1,

    and fold with probability, 0

                                          25
 The value of this pair of strategies is, ---, and in decmimals, 0.08503401361
                                          294

                  --------------------------------------------

                  --------------------------------------------

           If the deck has, 8, distinct cards and the bet size is, 1



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 2/3,

    and check with probability, 1/3

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 7,  he should bet With probability, 1,

    and check with probability, 0

If Player 1  got card, 8,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 2/3,

    and fold with probability, 1/3

If Player 2 got card, 5,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 6,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 7,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 8,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 2/21, and in decmimals, 0.09523809524

                  --------------------------------------------

           If the deck has, 8, distinct cards and the bet size is, 2



                         There are, 3, Nash equilibria

                                 here they are

                                  NE number, 1

Player 1 bets if his card is in the set, {1, 7, 8},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {4, 7, 8}, and folds otherwies , the value of the game is, 3/28

                                  NE number, 2

Player 1 bets if his card is in the set, {1, 7, 8},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {5, 7, 8}, and folds otherwies , the value of the game is, 3/28

                                  NE number, 3

Player 1 bets if his card is in the set, {1, 7, 8},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {6, 7, 8}, and folds otherwies , the value of the game is, 3/28

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 1,

    and check with probability, 0

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 7,  he should bet With probability, 1,

    and check with probability, 0

If Player 1  got card, 8,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 5,  he should call With probability, 1/2,

    and fold with probability, 1/2

If Player 2 got card, 6,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 7,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 8,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 3/28, and in decmimals, 0.1071428571

                  --------------------------------------------

           If the deck has, 8, distinct cards and the bet size is, 3



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 3/5,

    and check with probability, 2/5

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 7,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 8,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 5,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 6,  he should call With probability, 4/5,

    and fold with probability, 1/5

If Player 2 got card, 7,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 8,  he should call With probability, 1,

    and fold with probability, 0

                                          27
 The value of this pair of strategies is, ---, and in decmimals, 0.09642857143
                                          280

                  --------------------------------------------

           If the deck has, 8, distinct cards and the bet size is, 4



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 2/3,

    and check with probability, 1/3

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 7,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 8,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 5,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 6,  he should call With probability, 1/3,

    and fold with probability, 2/3

If Player 2 got card, 7,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 8,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 2/21, and in decmimals, 0.09523809524

                  --------------------------------------------

           If the deck has, 8, distinct cards and the bet size is, 5



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 5/7,

    and check with probability, 2/7

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 7,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 8,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 5,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 6,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 7,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 8,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 5/56, and in decmimals, 0.08928571429

                  --------------------------------------------

                  --------------------------------------------

           If the deck has, 9, distinct cards and the bet size is, 1



                         There are, 5, Nash equilibria

                                 here they are

                                  NE number, 1

Player 1 bets if his card is in the set, {1, 7, 8, 9},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {3, 5, 7, 8, 9}, and folds otherwies , the value of the game is, 7/72

                                  NE number, 2

Player 1 bets if his card is in the set, {1, 7, 8, 9},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {3, 6, 7, 8, 9}, and folds otherwies , the value of the game is, 7/72

                                  NE number, 3

Player 1 bets if his card is in the set, {1, 7, 8, 9},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {4, 5, 7, 8, 9}, and folds otherwies , the value of the game is, 7/72

                                  NE number, 4

Player 1 bets if his card is in the set, {1, 7, 8, 9},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {4, 6, 7, 8, 9}, and folds otherwies , the value of the game is, 7/72

                                  NE number, 5

Player 1 bets if his card is in the set, {1, 7, 8, 9},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {5, 6, 7, 8, 9}, and folds otherwies , the value of the game is, 7/72

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 1,

    and check with probability, 0

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 7,  he should bet With probability, 1,

    and check with probability, 0

If Player 1  got card, 8,  he should bet With probability, 1,

    and check with probability, 0

If Player 1  got card, 9,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 1/3,

    and fold with probability, 2/3

If Player 2 got card, 5,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 6,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 7,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 8,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 9,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 7/72, and in decmimals, 0.09722222222

                  --------------------------------------------

           If the deck has, 9, distinct cards and the bet size is, 2



                         There are, 7, Nash equilibria

                                 here they are

                                  NE number, 1

Player 1 bets if his card is in the set, {1, 8, 9},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {3, 6, 8, 9}, and folds otherwies , the value of the game is, 1/9

                                  NE number, 2

Player 1 bets if his card is in the set, {1, 8, 9},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {3, 7, 8, 9}, and folds otherwies , the value of the game is, 1/9

                                  NE number, 3

Player 1 bets if his card is in the set, {1, 8, 9},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {4, 6, 8, 9}, and folds otherwies , the value of the game is, 1/9

                                  NE number, 4

Player 1 bets if his card is in the set, {1, 8, 9},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {4, 7, 8, 9}, and folds otherwies , the value of the game is, 1/9

                                  NE number, 5

Player 1 bets if his card is in the set, {1, 8, 9},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {5, 6, 8, 9}, and folds otherwies , the value of the game is, 1/9

                                  NE number, 6

Player 1 bets if his card is in the set, {1, 8, 9},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {5, 7, 8, 9}, and folds otherwies , the value of the game is, 1/9

                                  NE number, 7

Player 1 bets if his card is in the set, {1, 8, 9},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {6, 7, 8, 9}, and folds otherwies , the value of the game is, 1/9

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 1,

    and check with probability, 0

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 7,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 8,  he should bet With probability, 1,

    and check with probability, 0

If Player 1  got card, 9,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 5,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 6,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 7,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 8,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 9,  he should call With probability, 1,

    and fold with probability, 0

  The value of this pair of strategies is, 1/9, and in decmimals, 0.1111111111

                  --------------------------------------------

           If the deck has, 9, distinct cards and the bet size is, 3



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 1,

    and check with probability, 0

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 7,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 8,  he should bet With probability, 2/3,

    and check with probability, 1/3

If Player 1  got card, 9,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 5,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 6,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 7,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 8,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 9,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 7/72, and in decmimals, 0.09722222222

                  --------------------------------------------

           If the deck has, 9, distinct cards and the bet size is, 4



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 2/3,

    and check with probability, 1/3

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 7,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 8,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 9,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 5,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 6,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 7,  he should call With probability, 2/3,

    and fold with probability, 1/3

If Player 2 got card, 8,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 9,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 5/54, and in decmimals, 0.09259259259

                  --------------------------------------------

           If the deck has, 9, distinct cards and the bet size is, 5



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 5/7,

    and check with probability, 2/7

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 7,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 8,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 9,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 5,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 6,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 7,  he should call With probability, 2/7,

    and fold with probability, 5/7

If Player 2 got card, 8,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 9,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 5/56, and in decmimals, 0.08928571429

                  --------------------------------------------

                  --------------------------------------------

           If the deck has, 10, distinct cards and the bet size is, 1



                         There are, 5, Nash equilibria

                                 here they are

                                  NE number, 1

Player 1 bets if his card is in the set, {1, 8, 9, 10},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {3, 5, 7, 8, 9, 10}, and folds otherwies , the value of the game is, 1/10

                                  NE number, 2

Player 1 bets if his card is in the set, {1, 8, 9, 10},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {3, 6, 7, 8, 9, 10}, and folds otherwies , the value of the game is, 1/10

                                  NE number, 3

Player 1 bets if his card is in the set, {1, 8, 9, 10},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {4, 5, 7, 8, 9, 10}, and folds otherwies , the value of the game is, 1/10

                                  NE number, 4

Player 1 bets if his card is in the set, {1, 8, 9, 10},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {4, 6, 7, 8, 9, 10}, and folds otherwies , the value of the game is, 1/10

                                  NE number, 5

Player 1 bets if his card is in the set, {1, 8, 9, 10},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {5, 6, 7, 8, 9, 10}, and folds otherwies , the value of the game is, 1/10

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 1,

    and check with probability, 0

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 7,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 8,  he should bet With probability, 1,

    and check with probability, 0

If Player 1  got card, 9,  he should bet With probability, 1,

    and check with probability, 0

If Player 1  got card, 10,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 5,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 6,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 7,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 8,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 9,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 10,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 1/10, and in decmimals, 0.1000000000

                  --------------------------------------------

           If the deck has, 10, distinct cards and the bet size is, 2



                         There are, 7, Nash equilibria

                                 here they are

                                  NE number, 1

Player 1 bets if his card is in the set, {1, 9, 10},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {4, 7, 9, 10}, and folds otherwies , the value of the game is, 1/9

                                  NE number, 2

Player 1 bets if his card is in the set, {1, 9, 10},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {4, 8, 9, 10}, and folds otherwies , the value of the game is, 1/9

                                  NE number, 3

Player 1 bets if his card is in the set, {1, 9, 10},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {5, 7, 9, 10}, and folds otherwies , the value of the game is, 1/9

                                  NE number, 4

Player 1 bets if his card is in the set, {1, 9, 10},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {5, 8, 9, 10}, and folds otherwies , the value of the game is, 1/9

                                  NE number, 5

Player 1 bets if his card is in the set, {1, 9, 10},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {6, 7, 9, 10}, and folds otherwies , the value of the game is, 1/9

                                  NE number, 6

Player 1 bets if his card is in the set, {1, 9, 10},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {6, 8, 9, 10}, and folds otherwies , the value of the game is, 1/9

                                  NE number, 7

Player 1 bets if his card is in the set, {1, 9, 10},

    and checks otherwise, while Player 2 calls if his card is in the set,

    {7, 8, 9, 10}, and folds otherwies , the value of the game is, 1/9

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 1,

    and check with probability, 0

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 7,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 8,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 9,  he should bet With probability, 1,

    and check with probability, 0

If Player 1  got card, 10,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 5,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 6,  he should call With probability, 1/2,

    and fold with probability, 1/2

If Player 2 got card, 7,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 8,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 9,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 10,  he should call With probability, 1,

    and fold with probability, 0

  The value of this pair of strategies is, 1/9, and in decmimals, 0.1111111111

                  --------------------------------------------

           If the deck has, 10, distinct cards and the bet size is, 3



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 1,

    and check with probability, 0

If Player 1  got card, 2,  he should bet With probability, 1/5,

    and check with probability, 4/5

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 7,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 8,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 9,  he should bet With probability, 1,

    and check with probability, 0

If Player 1  got card, 10,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 5,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 6,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 7,  he should call With probability, 1/5,

    and fold with probability, 4/5

If Player 2 got card, 8,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 9,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 10,  he should call With probability, 1,

    and fold with probability, 0

                                           23
  The value of this pair of strategies is, ---, and in decmimals, 0.1022222222
                                           225

                  --------------------------------------------

           If the deck has, 10, distinct cards and the bet size is, 4



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 2/3,

    and check with probability, 1/3

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 7,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 8,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 9,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 10,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 5,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 6,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 7,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 8,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 9,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 10,  he should call With probability, 1,

    and fold with probability, 0

 The value of this pair of strategies is, 4/45, and in decmimals, 0.08888888889

                  --------------------------------------------

           If the deck has, 10, distinct cards and the bet size is, 5



                       There are no pure Nash Equilibria

                     Let's find ONE Mixed Nash Equilibrium

                             Player 1's strategy is

If Player 1  got card, 1,  he should bet With probability, 5/7,

    and check with probability, 2/7

If Player 1  got card, 2,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 3,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 4,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 5,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 6,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 7,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 8,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 9,  he should bet With probability, 0,

    and check with probability, 1

If Player 1  got card, 10,  he should bet With probability, 1,

    and check with probability, 0

                             Player 2's strategy is

If Player 2 got card, 1,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 2,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 3,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 4,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 5,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 6,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 7,  he should call With probability, 0,

    and fold with probability, 1

If Player 2 got card, 8,  he should call With probability, 4/7,

    and fold with probability, 3/7

If Player 2 got card, 9,  he should call With probability, 1,

    and fold with probability, 0

If Player 2 got card, 10,  he should call With probability, 1,

    and fold with probability, 0

                                          11
 The value of this pair of strategies is, ---, and in decmimals, 0.08730158730
                                          126



                               -----------------



              This ends this paper that took, 1226.604, seconds.