Tuesday, November 25, 2008

Bail-out game theory

Last week Shachar Kariv mentioned something interesting in 201.

Anyway, it was just a pseudo-serious reference to the so-called inconsistent bailout policies of the Fed (facilitate buyout of Bear Stearns, rescue AIG, let Lehman go bankrupt, etc.) but he mentioned that there might be a mixed strategy equilibrium in the bank-Fed game, and that the inconsistencies are probabilistically chosen outcomes. We could have a payoff structure something like this:
Bank / Fed Bail-out Don't Bail-out
Responsible 4,4 3,5
Irresponsible 5,3 2,2
Essentially, banks have a choice between responsible practices, which pay off moderately, and irresponsible practices, which pay off more most of the time but occasionally lead to disaster. Thus a bail-out helps responsible firms, but helps irresponsible firms much more. The Fed, which maximizes the utility of society, prefers for banks to be responsible, but the fall-out of a disaster when a bank is irresponsible is large.
There are three Nash equilibria in this setup. If banks are always irresponsible, the Fed will always bail them out. On the other hand, if the Fed never bails out the banks, the banks will behave responsibly.
Or, the bank and Fed can randomize their response. If the Fed bails out banks half the time, and banks are responsible half the time, both are playing their optimal strategy given the other player’s strategy.
So, if half the banks are responsible, the Fed’s inconsistent response could just be their mixed strategy response.
Of course this is an insanely simplified model. In reality, big banks hurt society much more when they fail than small banks, and the Fed can observe the systemic risk a particular bank failure poses, the banks choose policies before the Fed can commit to a bail-out policy, and the Fed can choose individual bail-out policies for different types of banks. And, more hypothetically speaking, the Fed can commit to policies that otherwise might be viewed as non-credible threats, by passing laws on what it can and can’t do. Really the complications are endless. But I thought it was a cute idea.