# Classification of Two-Person Games

I'm looking for a classification of all simultaneous, complete information, two-person, two-action games. By classification, I mean a partition of all payoff matrices into a (finite number) of categories. Sample categories would be prisoner's dilemma and Stag hunt.

I'm willing to accept substitutes, like a classification of a different set of games. There is no unique answer to this question, and I'm not looking for an answer in the context of a particular class.

A paper reference is fine, although an online source would be better.

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• I voted to close because at this stage i believe the question is too broad. How about the partition between games in which one of the players has a utility of 1 for some combination of actions, and games for which player never have a utility of 1 for any combination of action? I guess you see my point : there are just too many of these partitions and it is not clear what you are exactly looking for. – Martin Van der Linden Apr 22 '16 at 19:28
• A classification can be found in this book, but I don't like it very much. – Michael Greinecker Apr 24 '16 at 2:32

If you look at generic games, that is games where players are not indifferent between any choices such a classification is easy to produce. Let the strategy sets be $$S_1 = \left\{a,b\right\}, S_2 = \left\{c,d\right\}$$ and let the utility/payoff functions be denoted by $U_1$ and $U_2$.
The characterization will be based on the dynamics of the game, that is given a strategy profile $(s_1,s_2)$ will any player wish to alther his strategy. This will also give you all the equilibria. Without loss of generality you can assume $$U_1(a,c) > U_1(b,c).$$ (Should this not hold, switch the labels of the strategies $a$ and $b$.)
We still need to know if player 1 prefers $(a,d)$ to $(b,d)$ or vice versa, and also if player 2 prefers $(a,c)$ to $(a,d)$ and if he prefers $(b,c)$ to $(b,d)$. This gives eight possible combinations which will characterize the game classes. (But due to symmetry actually there will only be four classes.)

Category 1.
$$U_1(a,d) > U_1(b,d) \mbox{ and } U_2(a,c) > U_2(a,d) \mbox{ and } U_2(b,c) > U_2(b,d).$$ In this category $a$ and $c$ are strictly dominant strategies. A game like this will play out like the prisoners dilemma in that it has a single Nash-equilibrium which is also a dominant equilibrium.

Category 2.
$$U_1(a,d) < U_1(b,d) \mbox{ and } U_2(a,c) < U_2(a,d) \mbox{ and } U_2(b,c) > U_2(b,d).$$ This game will play out the matching pennies game, in that there are no equilibria.

Category 3.
$$U_1(a,d) < U_1(b,d) \mbox{ and } U_2(a,c) > U_2(a,d) \mbox{ and } U_2(b,c) < U_2(b,d).$$ This game will play out like a coordination game, e.g. Battle of the Sexes or Chicken. There are two equilibria, $(a,c)$ and $(b,d)$.

Category 4.
$$U_1(a,d) > U_1(b,d) \mbox{ and } U_2(a,c) > U_2(a,d) \mbox{ and } U_2(b,c) < U_2(b,d).$$ In this category $a$ is again strictly dominant. $c$ is not, but a rational player 1 will always play $a$, and $c$ is best response to that. I don't know if this game has a well accepted name, it has one Nash-equilibrium $(a,c)$.

Due to symmetry, all other combinations belong to one of these categories.

If you wish to admit indifference between several outcomes, it is still possible to use this kind of classification, but the number of categories will be much larger.

I don't have my copy here but I believe that the book A theory of moves by Steven Brams solves all possible games of the type you describe.