“Common Knowledge and Equilibria Switching,” N.A. Dalkiran & M. Hoffman (2011)

It is not unusual that, at 2 A.M. on any given Saturday morning, a less-than-forthright gentlemen will ask his acquaintance whether “she would like to come up for some coffee.” To those just learning game theory, there is something strange here. Both parties are fully aware that no coffee will be served at such a late hour. We are both fully capable of translating the innuendo into its real meaning: there is no uncertainty here. But why, then, will nobody just ask for sex? And how is this question related to financial crises?

But perhaps these situations are not that strange. We all know from Rubinstein’s Electronic Mail game (you may know this as the story of the two coordinating generals) that mutual knowledge is not common knowledge. Imagine two generals on different ridges are planning an attack, and the attack will only succeed if both parties get a “good” signal; if either of us draws a bad signal, we know the attack will fail. The generals can communicate with each other by a messenger on horseback, but with probability epsilon close to zero, the messenger falls off his horse and never delivers the message. When I send a horse out, I know my signal and that’s it. When I receive the first horsemen, I know the other general’s signal and my own. When he receives a message back, he knows his signal, he knows my signal, and he knows that I know his signal. And so on. After two horsemen, we both know the other got a good signal, but we do not know that the other person knows we know this. So “almost” common knowledge is not almost at all, since common knowledge requires the “I know that he knows that I know” chain to continue infinitely, and that will happen with probability zero. Similar “contagion” arguments have been explored by many others (writeups on similar papers by Morris, Rob and Shin and Weinstein and Yildiz can be found on this site).

Dalkiran and Hoffman explore a similar question: when do similar tricky issues concerning higher order knowledge lead to “switching” of equilibria? More precisely, consider a two player, two action game, where (A,A) and (B,B) are the only pure strategy Nash equilibria: in other words, a coordination game. Let one equilibrium be a high payoff equilibrium, and the other be a low payoff equilibrium. Let there be a number of states of the world, with each agent endowed with an information partition in the standard way. Does there exist an equilibrium set of strategies where (A,A) is played with probability 1 in at least one state, and (B,B) with probability 1 in another state? That is, what conditions on priors, payoffs and the information partitions allow for equilibrium strategies where the “focal point” varies in different states even when the payoff matrix is not state-dependent. And what that might tell us about “customs” or behavior like the “would you like to come up for a drink” scenario? (Trivially, of course, such an equilibrium exists if we can both identify state 1 and state 2 with probability 1; the interesting situations are those where our knowledge of the current state is imperfect and heterogeneous, though I hope you’ll agree that such a situation is the most natural one!)

The authors provide necessary and sufficient conditions for arbitrary games, but the following example they give works nicely; the exact conditions rely on definitions of evident events and common p-belief and other such technical terms which will be familiar to decision theorists but are a bit too tricky to explain to a general audience in this blog post – if you read this paper and want to know more about those concepts, Aumann’s 2-part “Interactive Epistemology” articles and Larry Samuelson’s 2004 JEL are good places to start.

Imagine one agent (Aygun, in their example) is a bouncer at a whorehouse, and another agent (Moshe – the authors have a footnote explaining that they use their own names in this disreputable example so as not to defame the good name of readers with common game theory named like Ann and Bob!) is an occasional john. Aygun sometimes reads and doesn’t notice who walks in the brothel, and Moshe occasionally looks at the ground and doesn’t notice whether the bouncer sees him. It is a social convention that people should not have close friendships with anyone if it is common knowledge that they attend a brothel. There are then two coordinating equilibria: (A,A) for future close friendships and (B,B) for future weak friendships, which are coordinating in the sense that unequal friendships are worth less than equal friendships for both parties. There are then five states: H, (R,G), (R’,G), (R,G’) and (R’,G’), where H is the state in which Moshe stays home, (R,G) is the state where Moshe goes to the brothel, he looks at the Ground, and Aygun Reads, (R’,G) is the state where Moshe goes to the brothel, he looks at the Ground, and Aygun does not Read, etc. Both Moshe and Aygun have a common prior about the probability of looking at the ground, of staying home, and of reading.

The interesting potential equilibria here is the one where agents play (A,A) in state H and play (B,B) in state (R’,G’), the state where eye contact is made at the brothel. In such an equilibrium, would Moshe do better to avoid eye contact, meaning that (A,A) is the equilibrium strategy in states (R,G) and (R’,G)? Using the main theorem of the paper, a simple sufficiency condition obtains, which essentially says that the interesting equilibria exists if Aygun reads with sufficiently high probability, and that Aygun does not expect Moshe to be at the brothel with sufficiently high probability given that he is reading. If those conditions hold, then when Moshe looks at the ground, he will reason that Aygun is likely to be reading, and since Aygun is likely to be reading, he is likely to believe Moshe is at home, and therefore Moshe expects that Aygun expects that Moshe will play A, hence Moshe expects Aygun will play A, hence Moshe plays A. And Aygun reasons in exactly the same manner, so (A,A) is played in all states where eye contact is not made. But remember what is going on in (R’,G), the state where Aygun is not reading and Moshe is looking at the ground. Aygun knows Moshe is going to the brothel because he sees him, and Moshe of course knows that he himself is going to the brothel. So there is mutual knowledge here, but not common knowledge. And yet moving from mutual to common knowledge will break the “good” payoffs!

Now it goes without saying that in these types of coordination games, there are always equilibria where either (A,A) is played in every state or (B,B) in every state. But to the extent that certain states are associated with certain “focal points”, the ways in which customs or focal points can or can’t change equilibria across states are totally non-trivial in situations where agents have different information partitions. For instance, the authors give an example of the focal point at a traffic light where the color of the light is obscured to the drivers with some probability. They also generate a simple model of a bank run where switching depends on how much we expect other people to be following the news. Given the importance of discontinuous jumps and expectations to the financial world, I don’t doubt that understanding how and why equilibria switch is supremely relevant to understanding how stable or fragile a given financial regime is. Who knew politely asking a girl up to your apartment after a date was so related to the stability of the international financial system!

http://www.kellogg.northwestern.edu/faculty/dalkiran/dalkiran-jmp.pdf (November 2011 working paper – this paper is the job market paper of N. Aygun Dalkiran, a colleague of mine at Kellogg MEDS. If your department is looking for a good theorist, give him a call!)

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