Two players A and B engage in a penny pitching game Player A

Two players A and B engage in a \"penny pitching\" game. Player A starts with a pennies and player B starts with b pennies. On each round of the game both players toss one of their coins. If the up faces of the two coins match A wins, if the up faces don\'t match B wins. The winner claims both tossed coins. The game proceeds until one player is out of coins. Assume each toss is done fairly. For i greaterthanorequalto 0 and j greaterthanorequalto 0, let the ordered pair (i, j) stand for the condition that A has i coins and B has j coins. Let P_i, j = the probability that A wins the game starting from (i, j) with P_i, 0 =1 What is the relationship between P_i, j and P_j, i. What is P_J, J? For i greaterthanorequalto 2, develop a recursion for P_i, j in terms of P_i-1, j+1 and P_i-2, j+2. For 0 lessthanorequalto j lessthanorequalto m, use the recursion from C. to solve for P_m+1+j, m-j in terms of P_m+1, m. Use the result of D and P_2m+1, 0 = 1 to obtain an explicit solution for P_i, j when i+j is odd. For 1 lessthanorequalto j lessthanorequalto m, use the recursion from C. to solve for P_m+j, m-j in terms of P_m+1, m-1. Use the result of F and P_2m, 0 = 1 to obtain an explicit solution for P_i, j when i+j is even. Any game in which each player\'s expected profit is zero is said to be fair. Determine if this game is fair.

Solution

A.)

P_i,j means probability that A wins the game starting from (i,j) with P_i,0=1

   P_i,j means probability that A wins the game starting from (j,i) with Pj_,0=1

B.)

  P_j,j means probability that A wins the game starting from (j,j) with P_i,0=1 such that i=j

C.)

For, i+j=constant,we have

If A will loose one coin ,then coin left with him is i-1,similarly now B has j+1 coins.

P_i,j=P_i-1,j+1+P_i,0

_and

P_i,j=P_i-2,j+2+2.P_i,0

D.)

As in part C,we have

  P_m+j+1,m-j+j.P_i,0=P_m+1,m

E.)

  P_i,j+P_i-1,j+P_i-1,j-1+P_i-2,j-1+....+P_2m+1,0=1

G.)

  P_i,j+P_i-1,j+P_i-1,j-1+P_i-2,j-1+....+P_2m,0=1