Just started an introductory course in game theory, and here is a problem we have been talking about.
So here is the description of the game.
Two players, each player starts by placing \$1 each into the pot. They each then roll a single 6 sided dice. Player 1 (knowing the value of his roll) then has the option to either add \$1 to the pot, or pass (in which case the player with the higher roll gets the pot). If player 1 added \$1 to the pot, Player 2 can then either call (bringing the total to \$4, and highest roll gets the pot), or fold (Player 1 gets the pot). A tie results in the pot being split.
So we started off by by looking at pure strategies (such as P1 only betting on 5,6) and how P2 can respond with strategies that will always beat P1. We moved on to looking at more optimal strategies (such as P1 betting on 1,5,6 and therefore bluffing).
What I was wondering, is how would one go about finding the optimal strategy? I figured you could (in theory) create every payout matrix, and find the one that had the highest payout to P1 and lowest loss for P2? It just seems that I am over thinking this...
Thanks for the help!
It's a simple game.Three cases:1.P1 add and P2 call.2.P1 add and P2 fold.3.P1 pass. If P1 adds 1 to the pot, P2 either call or fold. If P2 calls, the expected profit for both of them is 1/6 * 0+15/36 * 2+15/36 * (-2)=0. Even if P2 has the chance to add more than 1(at this case P1 needs to call), the expected profit is always 0. If P1 makes P2 folds, then he can win 1. If P1 pass, then the expected profit for P1 and P2 is also 0. Therefore, the optimal strategy is that P1 should always add and bluff P2 to make the opponent to fold. P2 should always call.