January 28, 2001 4 October 2005 Eric Rasmusen notes, erasmuse@indiana.edu http://rasmusen.org p. 174. G denotes being Good; B denotes being Bad. p. 175. Payoffs. If the individual is Good, and Nature choose Minus, then the crime does not occur, and so Society's choice of P versus NP does not matter. Notice that Society never fails to detect the criminal act. The only reason it would not punish the act is that it detects "too often"-- sometimes it detects an act when the individual did not make a deliberate choice to commit the criminal act. If that happens too often, Society does not want to punish the act. Oddly, Society is allowed to prefer not punishing a deliberate crime to punishing it, because it is allowed that a<2. The individual's utility is just as high if the act occurs accidentally as if he does it deliberately (4 in both cases). This makes sense if it is, for example, underpaying his taxes. It does not fit the interpretation of society mistakenly punishing someone when the act did not in fact occur. p. 178. The Max2 Min1 solution is really just subgame perfection. Player 1 (society) moves first, and so gets to choose the punishment regime, g, and then player 2 chooses whether to be Bad or Good. Rubinstein is trying to show not just that his solution is a Nash equilibrium, but, in effect, that it is subgame perfect too, but in 1979 subgame perfection was not in common enough use (despite having been invented in 1965) that he thought of applying it. p. 178. You go to jail if the act occurs and is detected more than (alpha+ alpha_t) times per repetition in t repetitions. Recall that alpha is the probability that the act will occur even if the individual chooses Good. The optimal alpha_t falls with t, naturally enough-- if there are 1000 repetitions, then Society is pretty sure that the individual has chosen Bad if the average number of acts is even a little greater than alpha per repetition, because of the Law of Large Numbers. p. 179. The proof part b-c is showing that this punishment policy and being Good is an equilibrium. To do this, Rubinstein starts by noting that there will be only a finite number of false punishments if the individual is Good, so the limit of the average utility of Society will be the same as if there were no punishments at all, which is the first- best ideal. Most of the proof involves showing that being Good is optimal for the individual. To do that, he shows that the individual will only find it optimal to intentionally commit the act a finite number of times, which would make the limit of his average utility the same as if he *never* chose Bad. Start with a fixed, small divergence of the average payoff above the equilibrium level, and call that divergence amount epsilon. Then pick N large enough that alpha_N + 4/N