A year ago, a mathematician friend of mine commented that “as far as I can tell, nobody has published a paper that just outlines all the different types of 2x2 quadrant-game payoffs”, and they spent a weekend charting out the different payoff matrixes, and just meditating on how they felt about each one, and how it fit into their various game theory intuitions. But, they didn’t get around to publishing it AFAICT.
This seems like a really obvious thing to do, but prior to this post I don’t think anyone had written it up publicly. (if someone does know of an existing article, feel free to link here). But regardless I think a good writeup of this is useful to have in the LessWrong body of knowledge.
Real life is obviously more complicated than 2x2 payoff games, but having a set of crisp formulations is helpful for orientation on more complex issues. And, the fact that many people default to using prisoner’s dilemma all the time even when it’s not really appropriate seems like an actual problem that needed fixing.
I have some sense that the pedagogy of this post could be improved. I’d previously commented that using different symbols would be helpful for me. I have a nagging sense that there is other more useful feedback I could give on how to articulate some of the games, but don’t have clear examples.
Those concerns are relatively minor though. Overall, thanks for the great post. :)
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=(aij) and B=(bij) are the payoff matrices for Alice and Bob, so the payoff to Alice’s strategy si against Bob’s strategy tj is aij for Alice and bij 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:
aij>akl if and only if cij>ckl
and
bij>bkl if and only if dij>dkl
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.
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 a11>a21 (3 > 1) but c11<c21 (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.)
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.
Curated.
A year ago, a mathematician friend of mine commented that “as far as I can tell, nobody has published a paper that just outlines all the different types of 2x2 quadrant-game payoffs”, and they spent a weekend charting out the different payoff matrixes, and just meditating on how they felt about each one, and how it fit into their various game theory intuitions. But, they didn’t get around to publishing it AFAICT.
This seems like a really obvious thing to do, but prior to this post I don’t think anyone had written it up publicly. (if someone does know of an existing article, feel free to link here). But regardless I think a good writeup of this is useful to have in the LessWrong body of knowledge.
Real life is obviously more complicated than 2x2 payoff games, but having a set of crisp formulations is helpful for orientation on more complex issues. And, the fact that many people default to using prisoner’s dilemma all the time even when it’s not really appropriate seems like an actual problem that needed fixing.
I have some sense that the pedagogy of this post could be improved. I’d previously commented that using different symbols would be helpful for me. I have a nagging sense that there is other more useful feedback I could give on how to articulate some of the games, but don’t have clear examples.
Those concerns are relatively minor though. Overall, thanks for the great post. :)
Herbert Gintis’s Game Theory Evolving (2nd edition) offers the following exercise. (Bolding and hyperlinks mine.)
Thanks for finding this! I’m a bit confused, though; it suggests that the game with payoffs
(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
and these aren’t equivalent. We have a11>a21 (3 > 1) but c11<c21 (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.)
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.