Zack_M_Davis comments on Classifying games like the Prisoner’s Dilemma

[Zack_M_Davis]

if someone does know of an existing article

Herbert Gintis’s Game Theory Evolving (2nd edition) offers the following exercise. (Bolding and hyperlinks mine.)

6.15 Characterizing 2 x 2 Normal Form Games I

We say a normal form game is generic if no two payoffs for the same player are equal. Suppose $A=\left(a_{ij}\right)$ and $B=\left(b_{ij}\right)$ are the payoff matrices for Alice and Bob, so the payoff to Alice’s strategy $s_i$ against Bob’s strategy $t_j$ is $a_{ij}$ for Alice and $b_{ij}$ for Bob. We say two generic 2 x 2 games with payoff matrices (A, B) and (C, D) are equivalent if, for all i, j, k, l = 1, 2:

$$a_{ij}>a_{kl}\text{if and only if}{c}_{ij}>c_{kl}$$

and

$$b_{ij}>b_{kl}\text{if and only if}{d}_{ij}>d_{kl}$$

In particular, if a constant is added to the payoffs to all the pure strategies of one player when played against a given pure strategy of the other player, the resulting game is equivalent to the original.

Show that equivalent 2 x 2 generic games have the same number of pure Nash equilibria and the same number of strictly mixed Nash equilibria. Show also that every generic 2 x 2 game is equivalent to either the prisoner’s dilemma (§3.11), the battle of the sexes (§3.9), or the hawk-dove (§3.10). Note that this list does not include throwing fingers (§3.8), which is not generic.

[philh]

Thanks for finding this! I’m a bit confused, though; it suggests that the game with payoffs

3,3   2,1
1,2   0,0

(an instance of Cake Eating), is equivalent to one of those named games. But… which? It only has one pure Nash equilibrium, so it can’t be either hawk-dove or BOS, which both have two. And it can’t be equivalent to PD—an instance of that would be

3,3   1,4
4,1   0,0

and these aren’t equivalent. We have $a_{11}>a_{21}$ (3 > 1) but $c_{11}<c_{21}$ (3 < 4). So what am I missing?

(I had intended to try look this up myself, but I’m unlikely to do that in a timely manner, so I’m just leaving a comment. No obligation on you, of course.)

[Ben]

My best guess is that the book considers cake-eating to be trivial (just both eat cake), and is therefore not worried about even thinking about it, so it slipped the list.