Given that there is significant existing variation in human intelligence,

it's curious that we are not all much smarter than we actually are.

Besides the well-known costs of higher intelligence (e.g., more energy

use, bigger heads causing more difficult births), it seems that being

smart can be a disadvantage when playing some non-zero-sum games. Here is

one example.

Consider an infinitely repeated game with 2n players, where in each round

all players are randomly matched against each other in n seperate

prisoner's dillema stage games. After each round is finished, the outcomes

are recorded and published.

One plausible outcome of this game is for everyone to follow this strategy

(let's call it A): Initially mark all players as "good". If anyone defects

against a player who is marked as "good", mark him as "bad". Play

"cooperate" against "good" players, "defect" against "bad" players.

Now suppose in each stage game, there is probability p that the outcome is

not made public. Also assume that n is large enough so that we can

disregard the possibility that two players might face each other again in

the future and remember a previous non-published outcome. Now depending on

p, the discount factor, and the actual payoffs, it can still be an

equilibrium for everyone to follow strategy A.

For example, suppose the payoffs are 2,2/3,-10/-10,3/0,0, and p=0.5. If a

player deviates from the above strategy and plays "defect" against a

"good" player, he gains 1 utility (compared to strategy A) for the current

round, but has a probability of 0.5 of losing 2 utility in each future

round.

Now further suppose that the random number generator used to decide

whether each outcome is published or not is only pseudorandom, and there

are some "smart" players who are able to recognize the pattern and predict

whether a given stage game's outcome will be published. And suppose it's

public knowledge who these "smart" players are. In this third game, its no

longer an equilibrium for everyone to follow strategy A, because a "smart"

player should always play "defect" in any round in which he predicts the

outcome won't be published. The "normal" players can follow strategy A, or

they can follow a modified strategy (B) which starts by marking all

"smart" players as "bad", in which case the "smart" players should also

start by marking all "normal" players as "bad".

In either case the total surplus is less than if there were no "smart"

players. But with some game parameters, only the latter is an equilibria,

in which case "smart" players actually end up worse off than "normal"

players. (Note that even when the first outcome is an equlibrium, it is

not coalition-proof. I.e., the "normal" players have an incentive to

collectively switch to strategy B.)

For example, consider the above payoffs again. When a "normal" player

faces a "smart" player, he knows there is .5 probability that the "smart"

player will defect. If he deviates from strategy A to play "defect", there

is .5 probability that he gains 10 utility, and .5 probability that he

gains 1 utility in the current round and loses no more than 2 utilities in

each future round. Therefore depending on the time discount factor he may

have an incentive to play "defect".

-Wei Dai