Folk Theorem
The Folk theorem says that, in an infinitely repeated version of the game, provided players are sufficiently patient, there is an equilibrium such that both players cooperate on the equilibrium path. But if the game is repeated a known finite number of times, in that case, both players will play the maximum defection Nash equilibrium in each period, i.e., they will defect each time.
Many results in theorems deal with achieving and maintaining a socially optimal equilibrium in repeated games. These results are collectively called ‘Folk Theorems’.
The set of equilibria in infinitely repeated games is gigantic, various combinations can exist apart from the Nash equilibrium depending on certain conditions like time, preference, etc. The Folk Theorem explains how certain equilibria can exist in infinitely repeated stage games that are different from the Nash equilibrium of maximum defection.
- Infinitely repeated games — A game with an infinite number of rounds is also equivalent (in terms of strategies to play) to a game in which the players in the game do not know for how many rounds the game is being played.
Infinite games (or games that are being repeated an unknown number of times) cannot be solved by backward induction as there is no “last round” to start the backward induction from.
Even if the game being played in each round is identical, repeating that game a finite or infinite number of times can, in general, lead to very different outcomes (equilibria), as well as very different optimal strategies.
- Grim Trigger — In game theory, a grim trigger (also called the grim strategy or just grim) is a trigger strategy or a repeated game.
Initially, a player using a grim trigger will cooperate, but as soon as the opponent defects (thus satisfying the trigger condition), the player using a grim trigger will defect for the remainder of the iterated game. Since a single defect by the opponent triggers defection forever, the grim trigger is the most strictly unforgiving of strategies in an iterated game.
- Discount Factor — Discount factors incorporate a few different ideas: time, the value of money, a person’s underlying impatient preferences, and an exogenous probability that the game might end before the next period. Thus, they provide a helpful mathematical solution while also maintaining empirical accuracy.
We solve the calculation problem by discounting each period’s payoff by δ, the Greek lowercase letter delta. It is a value between 0 and 1.
Take a Nash equilibrium stage game that is infinitely repeated and discount the future.
Consider any alternative set of strategies in the game apart from the Nash equilibrium in which the utility for those strategies is greater for each player than the payoff from the Nash equilibrium of the stage game.
For example, the prisoner’s dilemma presents a situation where two parties, separated and unable to communicate, must each choose between cooperating with the other or not. The highest reward for each party occurs when both parties choose to cooperate. In a prisoner’s dilemma, mutual cooperation works better and has a greater payoff for both players than the Nash equilibrium of mutual defection.
Continuing on the alternative strategies, if the discount factor is sufficiently high, a subgame perfect equilibria exists in which players incorporate and apply those alternative strategies as long as those strategies are generating a payoff strictly greater than the Nash equilibrium maximum defection payoff.
Exploring Grim Trigger further — playing the equilibrium of mutual cooperation and peace is a subgame perfect equilibrium which is started by playing an alternative strategy. If anybody deviates from that strategy we revert to Nash equilibrium.
The equation for Alternative strategy equilibria to exist and persist -
u(alt)/(1-δ) > u(max dev) + δu(NE)/(1-δ),
(Where u is the utility and NE= Nash Equilibrium)
As long as this equality holds for everyone oneyou can get alternative strategies to be played in equilibrium.
We can analyze evaluate from the equation that if the discount factor is sufficiently high then there is no profitable deviation for any player and thus the alternative strategy equilibria prevails.
The willingness of a player to continue alternative strategy payoffs depends on the payoff a player obtains from one shot deviation (maximum deviation) and the further payoff the player will extract after shifting to Nash equilibrium because of deviation by one player.
Consider a grim trigger game and its payoffs for players one and two
If the player takes the future into account, he will see that the payoff in long term is drastically reduced by deviating once as the long-term loss from a smaller payoff of Nash equilibrium is more than the profit from one-time defection.
On the other hand, players focusing on very short-term gains or players in finite repeated games, tend to defect to increase their payoff in the short term.
But if we consider a player defecting, the best defection a player can do is deviate and get a 4 payoff which is more than the mutual cooperation.
But what folk theorem tells us is that it does not have to be mutual cooperation as the alternative strategy, another strategy could involve one player cooperating 95% of the time and defecting 5% of the time while the other player cooperates all the time. If you do utility calculation for both the players, you find that both of them will still have a better payoff in infinitely repeated games than if a Nash equilibrium was established after defection which would result in lower payoffs for both the players in the long run.
This means that mutual cooperation isn’t the only equilibrium that allows us to break off from the Nash equilibrium.
This can be demonstrated through an example:
Take two major trading countries, China and the USA; both nations trade immensely with each other, and a lot of revenue is generated for both. U.S. goods and services trade with China totaled an estimated $615.2 billion in 2020.
There is no true effective way to enforce the trade agreements, but still, they are upheld for the most part because the relations are long-term and each country thinks of its benefits.
The USA has put trade restrictions and increased taxes on China several times, and even though China has often retaliated back, there has never been a complete defection, and a harsh widespread hammering down on mutual trade policies is yet to be seen.
This is because both of the countries know that the long-term profits that they gain from not retaliating wildly on small sanctions are still much more than both the nations pushing the grim trigger and imposing tie breaking constraints, as that would lead to large amounts of losses in trade revenue and it would not benefit either side.
It is also clear that this equilibrium exists because both of the countries give importance to the future too, and if a country is only focused on short-term competition, it would be harder to control and difficult to punish it in a credible way.
