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## Question from Past Microeconomics Qualifying Exam Edit

Spring 2005 - Section II, Question three, George Mason University

Two players play a Prisoners' Dilemma game followed by a Coordination Game. In the Coordination game, BOTH players prefer (Left, Left) to (Right, Right). Neither player discounts the future; ß=1.

Consider the following candidate equilibrium: Both players cooperate in the first game. Then in the second game:

1. If both players cooperated in the first game, both play Left.
2. Otherwise, both players play Right.

Could this be a subgame perfect Nash equilibrium? Explain.

## Answer Edit

Yes, it could be a subgame perfect Nash equilibrium (SPNE) if the coordination game is simultaneous rather than sequential. If the coordination game is simultaneous, then if (Left, Left) in the second round confers a benefit large enough for both players such that ((Coop, Coop), (Left, Left)) is preferred to any other round 1 strategy followed by (Right, Right), then ((Coop, Coop), (Left, Left)) is a SPNE.

If the Coordination game is sequential, however, this does not hold. By backwards induction, (Left, Left) is the only SPNE for such a sequential coordination game considered by itself. Therefore, essentially, the SPNE to any game followed by a sequential coordination game where (Left, Left) is Pareto-Optimal must include (Left, Left) in the last round, and hence the final payoffs will simply be whatever the payoffs up to the point of the Coordination game nodes are plus the (Left, Left) payoffs. In this example, then, if the PD is simultaneous and the coordination game sequential, the fact that the sequential coordination game follows essentially just means that each player analyzes the game as they would the PD payoff matrix with all the payoffs in all the possible worlds increased by whatever the (Left, Left) payoffs of the sequential coordination game are, which will not change their decisions and hence will not make (Coop, Coop) sustainable (since (Don’t, Don’t) is the only NE in a PD game).

The fact that both players prefer (Left, Left) in the simultaneous coordination game does not imply that (Left, Left) will definitely be played, because each player only has an incentive to play left GIVEN that the other plays Left as well. In the simultaneous coordination game, as in all simultaneous games, all NE are SPNE. Since (Right, Right) is an NE as well, it is a credible threat, and hence the credible threat of (Right, Right) even though (Left, Left) is pareto-optimal is what makes (Coop, Coop) sustainable as part of an SPNE.

The relative payoffs matter as well. If the payoff for not cooperating in the prisoner's dilemma is a million times the difference between the Left Left payoff and the RighT Right payoff, then the coordination game will not be influenTial. Specifically, it is SPNE if

$U(Cooperate)+U(Left)>U(Don't Cooperate)+U(Right)$

The simultaneous-vs.-sequential issue is, for this question, a non-issue. In a coordination game, (L,L) and (R,R) are both PSNE; regardless of which cell gives higher payoffs, from either cell neither player can improve by switching, therefore each is an NE.

See the 2005 midterm answer key at www.bcaplan.com: http://www.gmu.edu/departments/economics/bcaplan/e812/mid05ans.htm It notes that for a sequential coordination game, " Pareto-inferior equilibria are not subgame perfect."

Weird. Look, the Pareto optimality of (L,L) makes it a focal point, but it doesn't make (R,R) non-Nash. I can't explain Caplan's answer (and, I'll note, neither does he), but it says in his notes and everywhere else you'll ever look: Subgame perfect Nash equilibrium simply means Nash play in each subgame. In the sequential version, the Coordination game is its own subgame. There are two PSNE in the coordination game; their Pareto superiority or inferiority is irrelevant for the fact that they are NE.

I mean... If Caplan's midterm key is right, it means that in a stand-alone Coordination game, a Pareto inferior PSNE is really not a NE, i.e., that Nash play requires that we choose the "good" equilibrium. That is nonsense. It's so obvious from his notes that we are reliant on focal points or common sense to get us into the good equilibrium, not Nash play.

In a multi-level game that is just one simultaneous game after another, and there are no other complications (e.g. if you play one particular strategy you won't have to play any other games), all sets of NE are SPNE. However, what SPNE really means in a multi-level game is that the strategies hold for each subgame, that they are what each player would do at every node in the game tree. In this example, after playing the PD, the player who makes the decision at the next node knows that if he chooses R, the other will choose R and if he chooses L, the other will choose L, as well (this is backward induction). Therefore, he will choose L and not R. Therefore, in such a sequential coordination game, (R,R) is not an SPNE, though it is a NE because GIVEN that the other player chooses R, the player in question would rather R, as well. The point is that in a sequential game order matters, and if both players prefer L, neither will choose R at the node immediately following the PD, no matter what has transpired before.

Certainly, Caplan's notes are probably not as systematic and explicit as they should be; he makes some points that taken alone seem to imply that NE in all games=SPNE, but he also gives examples like the entry game (mentioned in section VII of his 3rd and 4th week notes), where he says that "The two PSNE are (In, Accommodate) and (Out, Fight). But only the first is subgame perfect."

Okay, I understand now. I rather stupidly wasn't catching the distinction between "multi-level" and "sequential". So "sequential" means a game where one player moves first, and then the other moves after observing that move. Yes, in that circumstance, subgame perfect Nash play does require that they play (L,L). (But how trivial is a sequential coordination game? No wonder in the answer key Caplan seems so befuddled by the idea that anyone would think that he could be talking about a sequential coordination game.)