Here is a game of cards. A and B have two cards each – one green and one red. If A shows green and B shows green, A wins 5 – 0. If A shows green and B shows red, A loses 2 – 3. If A shows red and B shows green, A loses 0 – 5. If A shows red and B shows red, A wins 5 – 0. Here is the representation of the rules.
Looking carefully at the rule, you can conclude that the game is in A’s favour. But can A guarantee the maximum score, and how?
Here is the payoff matrix in the game theory format.
Before getting into the proper formulation, let’s check what happens if A plays only green. A might get a few wins early on, but once B figures out, she will play only red and win by 1 (2-3). On the other hand, if A plays only red, B will play green and win 5 (0-5).
A mixes up
Let A mixes up the play at a probability PAG for green (1 – PAG for red). If she aims to provide no incentive for B to show either green or red,
The payoff for B showing green = Payoff for B showing red
0 x PAG + 5 x (1-PAG) = 3 x PAG + 0 x (1-PAG)
5 – 5 PAG = 3PAG
PAG = 5/8 = 0.625
B mixes up
Naturally, B may respond by mixing her game, PBG for green. Using the same argument from B’s standpoint
The payoff for A showing green = Payoff for A showing red
5 x PBG + 2 x (1-PBG) = 0 x PBG + 5 x (1-PBG)
5PBG + 2 – 2PBG = 5 – 5PBG
PBG = 3/8 = 0.375
Equilibrium outcome
At these rates (PAG, PBG), the expected outcome for A is:
(5/8)(3/8)(5) + (5/8)(5/8)(2) + (3/8)(3/8)(0) + (3/8)(5/8)5 = 3.125
And the expected outcome for B is:
(5/8)(3/8)(0) + (5/8)(5/8)(3) + (3/8)(3/8)(5) + (3/8)(5/8)0 = 1.875
