I taught a series of classes on game theory over the last few weeks at Arché. And one of the things that has always puzzled me about game theory is that it seems so hard to reduce orthodox views in game theory to orthodox views in decision theory. The puzzle is easy enough to state. A fairly standard game theoretic treatment of Matching Pennies and Prisoners’ Dilemma involves the following two claims.
(1) In Matching Pennies, the uniquely rational solution involves each player playing a mixed strategy.
(2) In Prisoners’ Dilemma, the uniquely rational solution is for each player to defect.
Causal decision theory says denies that mixed strategies can ever be better than all of the pure strategies of which they are mixtures, at least for strategies that are mixtures of finitely many pure strategies. So a causal decision theorist wouldn’t accept (1). And evidential decision theory says that sometimes, for example when one is playing with someone who is likely to do what you do, it is rational to cooperate in Prisoners’ Dilemma. So it seems that orthodox game theorists are neither causal decision theorists nor evidential decision theorists.
So what are they then? For a while, I thought they were essentially ratificationists. And all the worse for them, I thought, since I think ratificationism is a bad idea. But now I think I was asking the wrong question. Or, more precisely, I was thinking of game theoretic views as being answers to the wrong question.
The first thing to note is that problems in decision theory have a very different structure to problems in game theory. In decision theory, we state what options are available to the agent, what states are epistemically possible and, and this is crucial, what the probabilities are of those states. Standard approaches to decision theory don’t get off the ground until we have the last of those in place.
In game theory, we typically state things differently. Unless nature is to make a move, we simply state what options are available to the players, and what plays are available to each of the actors, and of course what will happen given each combination of moves. We are told that the players are rational, and that this is common knowledge, but we aren’t given the probabilities of each move. Now it is true that you could regard each of the moves available to the other players as a possible state of the world. Indeed, I think it should be at least consistent to do that. But in general if you do that, you won’t be left with a solvable decision puzzle, since you need to say something about the probabilities of those states/decisions.
So what game theory really offers is a model for simultaneously solving for the probability of different choices being made, and for the rational action given those choices. Indeed, given a game between two players, A and B, we typically have to solve for six distinct ‘variables’.
- A’s probability that A will make various different choices.
- A’s probability that B will make various different choices.
- A’s choice.
- B’s probability that A will make various different choices.
- B’s probability that B will make various different choices.
- B’s choice.
The game theorists method for solving for these six variables is typically some form of reflective equilibrium. A solution is acceptable iff it meets a number of equilibrium constraints. We could ask about whether there should be quite so much focus on equilibrium analysis as we actually find in game theory textbooks (and journal articles), but it is clear that solving a complicated puzzle like this using reflective equilibrium analysis is hardly outside the realm of familiar philosophical approaches
Looked at this way, it seems that we should think of game theory really not as part of decision theory, but as much a part of epistemology. After all, what we’re trying to do here is solve for what rationality requires the players credences to be, given some relatively weak looking constraints. We also try to solve for their decisions given these credences, but it turns out that is an easy part of the analysis; all the work is in the epistemology. So it isn’t wrong to call this part of game theory ‘interactive epistemology’, as is often done.
What are the constraints on an equilibrium solution to a game? At least the following constraints seem plausible. All but the first are really equilibrium constraints; the first is somewhat of a foundational constraint. (Though note that since ‘rational’ here is analysed in terms of equilibria, even that constraint is something of an equilibrium constraint.)
- If there is a uniquely rational thing for one of the players to do, then both players must believe they will do it (with probability 1). More generally, if there is a unique rational credence for us to have, as theorists, about what A and B will do, the players must share those credences.
- 1 and 3, and 5 and 6, must be in equilibrium. In particular, if a player believes they will do something (with probability 1), then they will do it.
- 2 and 3, and 4 and 6, must be in equilibrium. A players decision must maximise expected utility given her credence distribution over the space of moves available to the other player.
That much seems relatively uncontroversial, assuming that we want to go along with the project of finding equilibria of the game. But those criteria alone are much too weak to get us near to game theoretic orthodoxy. After all, in Matching Pennies they are consistent with the following solution of the game.
- Each player believes, with probability 1, that they will play Heads.
- Each player’s credence that the other player will play heads is 0.5.
- Each player plays Heads.
Every player maximises expected utility given the other player’s expected move. Each player is correct about their own move. And each player treats the other player as being rational. So we have many aspects of an equilibrium solution. Yet we are a long way short of a Nash equilibrium of the game, since the outcome is one where one player deeply regrets their play. What could we do to strengthen the equilibrium conditions? Here are four proposals.
First, we could add a truth rule.
- Everything the players believe must be true. This puts constraints on 1, 2, 4 and 5.
This is a worthwhile enough constraint, albeit one considerably more externalist friendly than the constraints we usually use in decision theory. But it doesn’t rule out the ‘solution’ I described here, since everything the players believe is true.
Second, we could add a converse truth rule.
- If something is true in virtue of the players’ credences, then each player believes it.
This would rule out our ‘solution’. After all, neither player believes the other player will play Heads, but both players will in fact play Heads. But in a slightly different case, the converse truth rule won’t help.
- Each player believes, with probability 0.9, that they will play Heads.
- Each player’s credence that the other player will play heads is 0.5.
- Each player plays Heads.
Now nothing is guaranteed by the players’ beliefs about their own play. But we still don’t have a Nash equilibrium. We might wonder if this is really consistent with converse truth. I think this depends on how we interpret the first clause. If we think that the first clause must mean that each player will use a randomising device to make their choice, one that has a 0.9 chance of coming up heads, then converse truth would say that each player should believe that they will use such a device. And then the Principal Principle would say that each player should have credence 0.9 that the other player will play Heads, so this isn’t an equilibrium. But I think this is an overly metaphysical interpretation of the first clause. The players might just be uncertain about what they will play, not certain that they will use some particular chance device. So we need a stronger constraint.
Third, then, we could try a symmetry rule.
- Each player should have the same credences about what A will do, and each player should have the same credences about what B will do.
This will get us to Nash equilibrium. That is, the only solutions that are consistent with the above constraints, plus symmetry, are Nash equilibria of the original game. But what could possibly justify symmetry? Consider the following simple cooperative game.
Each player must pick either Heads or Tails. Each player gets a payoff of 1 if the picks are the same, and 0 if the picks are different.
What could justify the claim that each player should have the same credence that A will pick Heads? Surely A could have better insight into this! So symmetry seems like too strong a constraint, but without symmetry, I don’t see how solving for our six ‘variables’ will inevitably point to a Nash equilibrium of the original game.
Perhaps we could motivate symmetry by deriving it from something even stronger. This is our fourth and final constraint, called uniqueness.
- There is a unique rational credence function given any evidence set.
Assume also that players aren’t allowed, for whatever reason, to use knowledge not written in the game table. Assume further that there is common knowledge of rationality, as we usually assume. Now uniqueness will entail symmetry. And uniqueness, while controversial, is a well known philosophical theory. Moreover, symmetry plus the idea that we are simultaneously solving for the players’ beliefs and actions gets us the result that players always believe that a Nash equilibrium is being played. And the correctness condition on player beliefs means that rational players will always play Nash equilibria.
So we sort of have it, an argument from well-known (if not that widely endorsed) philosophical premises to the conclusion that when there is common knowledge of rationality, any game ends up in a Nash equilibrium.
Of course, we’ve used a premise that entails something way stronger. Uniqueness entails that any game has a unique rational equilibrium. That’s not, to put it mildly, something game theorists usually accept. The little coordination game I presented from a few paragraphs back is a game that sure looks like it has multiple equilibria! So I haven’t succeeded in deriving orthodox game theoretic conclusions from orthodox philosophical premises. But I think this epistemological tack is a better way to make game theory and philosophy look a little closer than they look if one starts thinking of game theorists as working on special (and specially important) cases of decision problems.