Folk theorem (game theory)

Preliminaries

Any Nash equilibrium payoff in a repeated game must satisfy two properties:

1. Individual rationality (IR): the payoff must weakly dominate the minmax payoff profile of the constituent stage game. I.e, the equilibrium payoff of each player must be at least as large as the minmax payoff of that player. This is because a player achieving less than his minmax payoff always has incentive to deviate by simply playing his minmax strategy at every history.

2. Feasibility: the payoff must be a convex combination of possible payoff profiles of the stage game. This is because the payoff in a repeated game is just a weighted average of payoffs in the basic games.

Folk theorems are partially converse claims: they say that, under certain conditions (are different in each folk theorem), every payoff that is both IR and feasible can be realized as a Nash equilibrium payoff profile in 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 don't 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 infimum of the limit of means. If game results in a path of outcomes x t , player i's utility is:

where u i 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 IR feasible payoff profile in the basic game is an equilibrium payoff profile in the repeated game.

The proof employs what is called 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 players switch to the strategy which minmaxes player i forever after. The one-stage gain from deviation contributes 0 to the total utility of the player. The utility of a deviating player cannot be higher than his minmax payoff. Hence all players stay on the intended path.

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.

Coalition subgame-perfect equilibria: An equilibrium is called a coalition Nash equilibrium if no coalition can gain from deviating. It is called a coalition subgame-perfect equilibria if no coalition can gain from deviating after any history. With the limit-of-means criterion, an outcome is attainable in coalition-Nash-equilibrium or in coalition-subgame-perfect-equilibrium, if-and-only-if it is Pareto efficient and weakly-coalition-individually-rational.

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 i (strictly) prefers the sequence of outcomes y t over the sequence x t , if:

For example, consider the sequences x = ( 0 , 0 , 0 , 0 , . . . ) and y = ( − 1 , 2 , 0 , 0 , . . . ) . According to the limit-of-means criterion, they are equivalent but according to the overtaking criterion, y is better than x . 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:

Strict stationary equilibria: A Nash equilibrium is called strict if each player strictly prefers the infinite sequence of outcomes attained in equilibrium, over any othe sequence he can deviate to. A Nash equilibrium is called stationary if the outcome is the same in each time-period. An outcome is attainable in strict-stationary-equilibrium if-and-only-if for every player the outcome is strictly better than the player's minimax outcome.

Strict stationary subgame-perfect equilibria: An outcome is attainable in strict-stationary-subgame-perfect-equilibrium, if for every player the outcome is strictly better than the player's minimax outcome (note that this is not an "if-and-only-if" result). To achieve subgame-perfect equilibrium with the overtaking criterion, it is required to punish not only the player that deviates from the agreement path, but also every player that does not cooperate in punishing the deviant.

The "stationary equilibrium" concept can be generalized to a "periodic equilibrium", in which a finite number of outcomes is repeated periodically, and the payoff in a period is the arithmetic mean of the payoffs in the outcomes. That mean payoff should be strictly above the minimax payoff.

Strict stationary coalition equilibria: With the overtaking criterion, if an outcome is attainable in coalition-Nash-equilibrium, then it is Pareto efficient and weakly-coalition-individually-rational. On the other hand, if it is Pareto efficient and strongly-coalition-individually-rational, strongly-coalition-individually-rational it can be attained in strict-stationary-coalition-equilibrium.

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:

U i = ( 1 − δ ) ∑ t ≥ 0 δ t u i ( x t ) ,

The discount factor indicates how patient the players are.

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 pure strategy profile with payoff profile x which strictly dominates the minmax payoff profile. One can define a Nash equilibrium with x 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

1 1 − δ ϵ .

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 x to be any IR feasible payoff profile; it only requires there exists an IR feasible payoff profile x, which strictly dominates the minmax payoff profile. Then, the folk theorem guarantees that it is possible to approach x in equilibrium to any desired precision (for every ε there exists a Nash equilibrium where the payoff profile is a distance ε away from x).

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 a player in an finitely repeated game is given by a simple arithmetic mean:

U i = 1 T ∑ t = 0 T u i ( h t )

A folk theorem for this case has the following additional requirement:

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:

1. In the first phase, the players alternate strategies in the required frequencies to approximate the desired payoff profile.
2. In the last phase, the players play the preferred equilibrium of each of the players in turn.

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:

Anthropology: in a community where all behavior is well known, and where members of the community know that they will continue to have to deal with each other, then any pattern of behavior (traditions, taboos, etc.) may be sustained by social norms so long as the individuals of the community are better off remaining in the community than they would be leaving the community (the minimax condition).

International politics: agreements between countries cannot be effectively enforced. They are kept, however, because relations between countries are long-term and countries can use "minimax strategies" against each other. This possibility often depends on the discount factor of the relevant countries. If a country is very impatient (pays little attention to future outcomes), then it may be difficult to punish it (or punish it in a credible way). As specific historic example, consider the relations between North Vietnam and the United States during the Vietnam war.

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:

Horizon – whether game is repeated Finitely or Infinitely many times.

Utilities – whether the utility of a player in the repeated game is assumed to be an arithmetic mean or a discounted sum.

Conditions on G (the basic game) – whether there are any technical conditions that should hold in the one-shot game in order for the theorem to work.

Conditions on x (the target payoff vector) – whether the theorem works for any IR and feasible payoff vector, or only on a subset of these vectors.

Equilibrium type – if all conditions are met, what kind of equilibrium is guaranteed by the theorem – Nash or Subgame-perfect?

Punishment type – what kind of punishment strategy is used to deter players from deviating?