Decoding Game Theory’s Folk Theorem

Last Updated : 19 Jan, 2024

The folk theorem, which came into existence in the late fifties became a widespread phenomenon in artificial intelligence and game theory. This article is all about exploring Game theory and Folk Theorem.

Game Theory

Game theory is an area of mathematics and economics that investigates strategic interactions between rational decision-makers. It provides a framework for examining situations in which the result of one person’s decision is influenced by the decisions of others.

Key concepts in Game Theory

Below are some of the important concepts in game theory that you need to be aware of before diving into a folk theorem.

Nash Equilibrium: Nash Equilibrium is a key concept in game theory, named after mathematician and Nobel winner John Nash. In a Nash Equilibrium, each participant (player) in a game picks their strategy while taking into account the plans of others, and no player has an incentive to unilaterally modify their approach. In other words, Nash Equilibrium is a set of strategies where no player has an incentive to unilaterally deviate from their chosen strategy given the strategies chosen by others.
Utility: Utility is generally defined as the payoff of individual actions which can be grouped under transferable and non-transferable. Utility in game theory refers to the measure of satisfaction, happiness, or preference that individuals associate with different possible outcomes or decisions. It represents how much a player values or benefits from a particular situation or set of choices.
Coalitional Game Theory: Coalitional game theory shifts the emphasis away from individual agents and onto the collective behaviour of groups or coalitions of actors. In this framework, the emphasis is on understanding how groups of players work together to achieve goals rather than evaluating individual strategies in isolation.
Subgame Perfect Nash Equilibrium: Subgame perfect Nash Equilibrium is a refined version of Nash Equilibrium in the context of extensive-form games. An extensive-form game describes a sequential decision-making process, which is generally represented as a tree.

Folk theorem

Folk theorems are a set of theorems that provide for the abundance of Nash equilibrium payoffs in a repeated game. The original folk theorem, though unpublished, used to consider the payoffs of all the Nash equilibriums present on infinite rounds of game. This was further modified by Friedman in 1971, who was concerned with the payoff of certain subgame perfect Nash equilibria of an infinitely round game, thus strengthening the Folk theorem by using a stronger equilibrium concept of Subgame Perfect Nash Equilibria rather than the Nash Equilibria.

Folk theorem says that:

“In an infinite play of game, both the players will remain patient and will be in equilibrium such that both of them will cooperate on the equilibrium path.”

In case the same is made finite, then both of them would be playing maximum defection Nash equilibrium, thus deviating from the equilibrium and defecting each other.

It provides multiple equilibriums, other than Nash equilibrium that exist in space considering different parameters, such as time and preference.

Dynamics of Folk Theorem

Let us take an example of n player stage game where each player has a finite number of actions to choose from, independent of the fact other player’s actions. The payoff in this case will be decided based on the cumulative individual choices that the player made.

Then the player’s utility will be denoted as u_i(x) where x is the collective choice that the player i would be trying to maximize.

Now repeating the above stage game, the player will be choosing his actions based on the actions of other players as well as those identified from the previous iterations thus following the deterministic principle. There can be multiple ways to reach to payout profile by the strategy profile, which refers to the individual actions.

Such a Nash equilibrium payoff profile must conform to two properties.

Individual rationality: The equilibrium payoff must always be at least as large as the minimax payoff, defaulting on which the player has the option to deviate by simply playing the minimax strategy.
Feasibility: Since the payoff in a repeated game is the weighted average of the payoff in the basic games, thus it should be a convex combination of possible payoff profiles of the basic games.

Note: Convex combination is the linear combination of points where all coefficients are non-negative and sum to 1.

Infinite play

Infinite play of games is nothing but a play of the game that many times such that the players are not aware of or we can say, they are not provided with the initial description as to how many rounds will be played.

We can’t reason backwards in time in case of infinite play, from the end of the round to determine the sequence of optimal actions because there is no last round in case of an infinite number of plays.

Repeating a game for an infinite number of times will lead to very different equilibriums as well as very different optimal strategies.

Grim Trigger

Grim trigger is a trigger strategy in a repeated game.

In a repeated game, the player opting grim trigger will cooperate for several rounds till the equilibrium is reached. Once the opponent triggers the defect, the player will follow the grim trigger for the remainder of the term thus defecting the opponent.

Here, a single action of the opponent had given rise to a grim trigger and thus this is the most unforgiving strategy in a game.

Discount Factor

Discount factor relies on multiple factors, such as time, the person’s impatient preferences and the probability that the game might end before the period.

Consider any set of strategies other than the Nash equilibrium in a particular game.

Let’s consider the case of the Prisoner’s dilemma.

In this case, the prisoners have the choice of either defecting or mutually cooperating. The highest reward comes in the case both cooperate and remain silent, whereas as per the Nash equilibrium, the parties chose to defect each other. Thus, cooperation has a greater payoff and works better than the one provided by the Nash equilibrium.

We can solve the calculation by discounting each period’s payoff by $\delta$ , which ranges from 0 to 1.

If the discount factor is sufficiently high, the parties involved would continue to operate on the alternative strategy as long as it generates a payoff greater than the one provided by the Nash equilibrium maximum defection strategy.

Remaining on the alternative strategy denotes a subgame perfect equilibrium. If anyone among them deviates from it, they would again be inclined towards the Nash equilibrium.

The equation for alternative strategy equilibria is given by:

$\frac{u(alt)}{(1 - \delta)} > u(max dev) + \frac{\delta u(NE)}{(1 - \delta)}$

where,

u is the utility
δ represents the discount factor
alt signifies the alternative strategy
madethe denotes maximum deviation
NE represents Nash equilibrium.

This provides an alternative strategy that can be employed in a game. As long as the condition holds, they will remain to be played in equilibrium.

The folk theorem says that it should not always be cooperation as the alternative strategy, there can even be another alternative where the person cooperates for 80% of the time, and let’s say, defects for the remaining 20% of the time. In such cases, we can see that the utility remains high as compared to the Nash equilibrium which can provide lower payoffs for both the players in the long run.

Infinitely-Repeated Games without Discounting

In the case of undiscounted models, since the players are not much concerned about the periods and hence don’t differentiate between the utilities, their utility in the repeated game is represented by the sum of the utilities in the basic games.

$U_i = \liminf_{T \to \infty} \frac{1}{T} \sum_{t=1}^{T} \delta \cdot u_i(x_t)$

where,

U_i represents the utility of the^th player
u_iUI is the basic game utility function
x_ttx denotes the outcomes with the collective choices of the players at iteration t
T being the repeated rounds in the game where $T \rightarrow \infty$

Folk theorem Example

Consider the economic trade dynamics of the two countries. To enforce trade agreements, both countries use sanctions, selectively imposing limitations on one another. The repetition of these behaviours is what makes them intriguing.

To enforce trade agreements, they opt for sanctions to be placed on each of them. They will continue placing such restrictions because it will be in the larger interest of the nations. But they will not go for complete reversal and break off keeping future considerations.

Looking forward to these small sanctions can still be continuing their larger interests in place of total rejection which will close the ties with the huge loss to both of them.

The Nash equilibrium would look forward to completing reversal, but the better alternative would be to continue with such small sanctions which won’t click the grim trigger and hence would be in the larger interest of maintaining the equilibrium.

Applications of Folk theorem

Folk theorem can be applied to many diverse fields. Below are some such fields of interest.

In the international arena jointly work on agreements between two countries, as provided in the example above.
In the social sector, where the individuals are aware of the fact that all the behaviours are well known and where they have to be as a single entity, so they will be following the minimax equilibrium.
In economic terms, where negotiation becomes the best method to opt for a purchase.

Conclusion

Folk theorem suggests that if the players are patient enough and far-sighted, then the repeated interaction can result in any feasible and rational average payoff in a Subgame Perfect Equilibrium. In such a scenario, the discount factor tends to be 1. Such feasibility is provided by the convex combination of possible payoff profiles in the basic game and the payoff dominating the min-max profile provides an individual rationality that must be satisfied for a repeated game Nash equilibrium.

Suggest improvement

Coalitional Game theory

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