GAME PROBLEM
At a tv game show, two players are handed six cards each. Each of the six cards shows a number, and both players have the same cards. The six numbers are $0,0,3,0,15$. The game is as follows, each one of the players should choose a card, independently of the other player. After this, both players show their selected card. If the numbers agree, both players receive certain amount of money equal to $M > 0$. Define the selection of a number as a strategy. Draw a normal form and find all Nash Equilibria, but one of these seems more plausible than the others. Which one?
I tried to write it in the following normal form:
I don't know it this is correct, also, I don't know how to find the Nash Equilibria, can anybody please help me?
Trivially, all strategy profiles that lead to payoffs $M$ for both are Nash equilibria. This is a coordination game, where players have no antagonistic interests at all. Both maximize their gains by cooperating, i.e., choosing the same cards.
How do you find Nash equilibria in normal form games? I explained a nice method here.
Alternatively, just guess that some strategy profile, say $(0,0)$, is a Nash equilibrium. This is your "equilibrium candidate". Now check if there is some player who strictly prefers to deviate, i.e., chooses another strategy. By definition, if there is no such a deviation, then your candidate is an equilibrium. We only have two players, so checking this can be quickly done. At $(0,0)$, if the column player deviates to "1", he reduces his payoff from $M>0$ to 0, so this is not a profitable deviation. Same for deviations to "3" or "15". Choosing another "0" card gives him $M$ as in the candidate, so there is no deviation that makes him strictly better off (gives him more than $M$). The same holds for the row player and any of the strategy profiles with $M$ for both. Hence, all strategy profiles where both get $M$ are Nash equilibria.