Class of theorems about Nash equilibrium payoff profiles in repeated games
In game theory, folk theorems are a class of theorems describing an abundance of Nash equilibrium payoff profiles in repeated games . The original Folk Theorem concerned the payoffs of all the Nash equilibria of an infinitely repeated game. This result was called the Folk Theorem because it was widely known among game theorists in the 1950s, even though no one had published it. Friedman's (1971) Theorem concerns the payoffs of certain subgame-perfect Nash equilibria (SPE) of an infinitely repeated game, and so strengthens the original Folk Theorem by using a stronger equilibrium concept: subgame-perfect Nash equilibria rather than Nash equilibria.
The Folk Theorem suggests that if the players are patient enough and far-sighted (i.e. if the discount factor formula_1), then repeated interaction can result in virtually any average payoff in an SPE equilibrium. "Virtually any" is here technically defined as "feasible" and "individually rational".
Setup and definitions.
We start with a "basic game", also known as the "stage game", which is an "n-"player game. In this game, each player has finitely many actions to choose from, and they make their choices simultaneously and without knowledge of the other player's choices. The collective choices of the players leads to a "payoff profile," i.e. to a payoff for each of the players. The mapping from collective choices to payoff profiles is known to the players, and each player aims to maximize their payoff. If the collective choice is denoted by "x," the payoff that player "i" receives, also known as player "i"'s "utility", will be denoted by formula_2.
We then consider a repetition of this stage game, finitely or infinitely many times. In each repetition, each player chooses one of their stage game options, and when making that choice, they may take into account the choices of the other players in the prior iterations. In this repeated game, a "strategy" for one of the players is a deterministic rule that specifies the player's choice in each iteration of the stage game, based on all other player's choices in the prior iterations. A choice of strategy for each of the players is a "strategy profile," and it leads to a payout profile for the repeated game. There are a number of different ways such a strategy profile can be translated into a payout profile, outlined below.
Any Nash equilibrium payoff profile of a repeated game must satisfy two properties:
Folk theorems are partially converse claims: they say that, under certain conditions (which are different in each folk theorem), "every" payoff profile that is both individually rational and feasible can be realized as a Nash equilibrium payoff profile of the repeated game.
There are various folk theorems; some relate to finitely-repeated games while others relate to infinitely-repeated games.
Infinitely-repeated games without discounting.
In the undiscounted model, the players are patient. They do not differentiate between utilities in different time periods. Hence, their utility in the repeated game is represented by the sum of utilities in the basic games.
When the game is infinite, a common model for the utility in the infinitely-repeated game is the limit inferior of mean utility: If the game results in a path of outcomes formula_3, where formula_3 denotes the collective choices of the players at iteration "t" ("t=0,1,2...)," player "i"'s utility is defined as
formula_5
where formula_6 is the basic-game utility function of player "i".
An infinitely-repeated game without discounting is often called a "supergame".
The folk theorem in this case is very simple and contains no pre-conditions: every individually rational and feasible payoff profile in the basic game is a Nash equilibrium payoff profile in the repeated game.
The proof employs what is called a "grim" or "grim trigger" strategy. All players start by playing the prescribed action and continue to do so until someone deviates. If player "i" deviates, all other players switch to picking the action which minmaxes player "i" forever after. The one-stage gain from deviation contributes 0 to the total utility of player "i". The utility of a deviating player cannot be higher than his minmax payoff. Hence all players stay on the intended path and this is indeed a Nash equilibrium.
Subgame perfection.
The above Nash equilibrium is not always subgame perfect. If punishment is costly for the punishers, the threat of punishment is not credible.
A subgame perfect equilibrium requires a slightly more complicated strategy. The punishment should not last forever; it should last only a finite time which is sufficient to wipe out the gains from deviation. After that, the other players should return to the equilibrium path.
The limit-of-means criterion ensures that any finite-time punishment has no effect on the final outcome. Hence, limited-time punishment is a subgame-perfect equilibrium.
Overtaking.
Some authors claim that the limit-of-means criterion is unrealistic, because it implies that utilities in any finite time-span contribute 0 to the total utility. However, if the utilities in any finite time-span contribute a positive value, and the value is undiscounted, then it is impossible to attribute a finite numeric utility to an infinite outcome sequence. A possible solution to this problem is that, instead of defining a numeric utility for each infinite outcome sequence, we just define the preference relation between two infinite sequences. We say that agent formula_7 (strictly) prefers the sequence of outcomes formula_8 over the sequence formula_3, if:
formula_10
For example, consider the sequences formula_11 and formula_12. According to the limit-of-means criterion, they provide the same utility to player "i," but according to the overtaking criterion, formula_13 is better than formula_14 for player "i". See overtaking criterion for more information.
The folk theorems with the overtaking criterion are slightly weaker than with the limit-of-means criterion. Only outcomes that are "strictly" individually rational, can be attained in Nash equilibrium. This is because, if an agent deviates, he gains in the short run, and this gain can be wiped out only if the punishment gives the deviator strictly less utility than the agreement path. The following folk theorems are known for the overtaking criterion:
Infinitely-repeated games with discounting.
Assume that the payoff of a player in an infinitely repeated game is given by the "average discounted criterion" with discount factor 0  ; "δ"  ; 1:
formula_15
The discount factor indicates how patient the players are. The factor formula_16 is introduced so that the payoff remain bounded when formula_17.
The folk theorem in this case requires that the payoff profile in the repeated game strictly dominates the minmax payoff profile (i.e., each player receives strictly more than the minmax payoff).
Let "a" be a strategy profile of the stage game with payoff profile "u" which strictly dominates the minmax payoff profile. One can define a Nash equilibrium of the game with "u" as resulting payoff profile as follows:
1. All players start by playing "a" and continue to play "a" if no deviation occurs.
2. If any one player, say player "i", deviated, play the strategy profile "m" which minmaxes "i" forever after.
3. Ignore multilateral deviations.
If player "i" gets "ε" more than his minmax payoff each stage by following 1, then the potential loss from punishment is
formula_18
If "δ" is close to 1, this outweighs any finite one-stage gain, making the strategy a Nash equilibrium.
An alternative statement of this folk theorem allows the equilibrium payoff profile "u" to be any individually rational feasible payoff profile; it only requires there exist an individually rational feasible payoff profile that strictly dominates the minmax payoff profile. Then, the folk theorem guarantees that it is possible to approach "u" in equilibrium to any desired precision (for every "ε" there exists a Nash equilibrium where the payoff profile is a distance "ε" away from "u").
Subgame perfection.
Attaining a subgame perfect equilibrium in discounted games is more difficult than in undiscounted games. The cost of punishment does not vanish (as with the limit-of-means criterion).
It is not always possible to punish the non-punishers endlessly (as with the overtaking criterion) since the discount factor makes punishments far away in the future irrelevant for the present. Hence, a different approach is needed: the punishers should be rewarded.
This requires an additional assumption, that the set of feasible payoff profiles is full dimensional and the min-max profile lies in its interior. The strategy is as follows.
1. All players start by playing "a" and continue to play "a" if no deviation occurs.
2. If any one player, say player "i", deviated, play the strategy profile "m" which minmaxes "i" for "N" periods. (Choose "N" and "δ" large enough so that no player has incentive to deviate from phase 1.)
3. If no players deviated from phase 2, all player "j" ≠ "i" gets rewarded "ε" above "j"'s min-max forever after, while player "i" continues receiving his min-max. (Full-dimensionality and the interior assumption is needed here.)
4. If player "j" deviated from phase 2, all players restart phase 2 with "j" as target.
5. Ignore multilateral deviations.
Player "j" ≠ "i" now has no incentive to deviate from the punishment phase 2. This proves the subgame perfect folk theorem.
Finitely-repeated games without discount.
Assume that the payoff of player "i" in a game that is repeated "T" times is given by a simple arithmetic mean:
formula_19
A folk theorem for this case has the following additional requirement:
In the basic game, for every player "i", there is a Nash-equilibrium formula_20 that is strictly better, for "i", than his minmax payoff.
This requirement is stronger than the requirement for discounted infinite games, which is in turn stronger than the requirement for undiscounted infinite games.
This requirement is needed because of the last step. In the last step, the only stable outcome is a Nash-equilibrium in the basic game. Suppose a player "i" gains nothing from the Nash equilibrium (since it gives him only his minmax payoff). Then, there is no way to punish that player.
On the other hand, if for every player there is a basic equilibrium which is strictly better than minmax, a repeated-game equilibrium can be constructed in two phases:
In the last phase, no player deviates since the actions are already a basic-game equilibrium. If an agent deviates in the first phase, he can be punished by minmaxing him in the last phase. If the game is sufficiently long, the effect of the last phase is negligible, so the equilibrium payoff approaches the desired profile.
Applications.
Folk theorems can be applied to a diverse number of fields. For example:
On the other hand, MIT economist Franklin Fisher has noted that the folk theorem is not a positive theory. In considering, for instance, oligopoly behavior, the folk theorem does not tell the economist what firms will do, but rather that cost and demand functions are not sufficient for a general theory of oligopoly, and the economists must include the context within which oligopolies operate in their theory.
In 2007, Borgs et al. proved that, despite the folk theorem, in the general case computing the Nash equilibria for repeated games is not easier than computing the Nash equilibria for one-shot finite games, a problem which lies in the PPAD complexity class. The practical consequence of this is that no efficient (polynomial-time) algorithm is known that computes the strategies required by folk theorems in the general case.
Summary of folk theorems.
The following table compares various folk theorems in several aspects:
Folk theorems in other settings.
In allusion to the folk theorems for repeated games, some authors have used the term "folk theorem" to refer to results on the set of possible equilibria or equilibrium payoffs in other settings, especially if the results are similar in what equilibrium payoffs they allow. For instance, Tennenholtz proves a "folk theorem" for program equilibrium. Many other folk theorems have been proved in settings with commitment.
Notes.
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