Repeated Games

Repeated prisoners' dilemma

The prisoners' dilemma is a paradox in the analysis of decisions where the optimum outcome is not produced by two individuals acting in their own interests. The typical dilemma of the prisoner is set up so that at the expense of the other participant both parties choose to protect themselves. The implication is that all parties are in a poorer condition than if they had cooperated in the decision-making phase. One of the most common ideas in contemporary game theory is the prisoner's dilemma.

Prisoner's Dilemma Examples

The economy is full of illustrations of the dilemmas posed by inmates, which may offer advantages or damage to the economy and culture in general. The most popular topic is scenarios in which the incentives each decision-maker experiences will allow each person to act in such a way that everyone aggravates them jointly and personally rejects choices that would allow everyone to mutually benefit if everyone could somehow choose cooperatively.

The tragedy of ordinary people is one such example. It could be a mutual interest for all to preserve and reinvest in a pool of natural resources and continue to use it, but each person still has an opportunity to use the resources as rapidly as possible instead. It would obviously make everything happier here if we could find a way to collaborate.

The behavior of cartels can also be regarded as a dilemma for prisoners. While all cartel members will collectively become more improved by a decrease in demand, holding the price one gets sufficiently to grab customers' economic rent, one cartel member has an opportunity for the cartel individually to steal and boost production such that rent is often taken away from the other cartel members. With respect to the general well being of the cartel, this shows how a prisoner's dilemma breaking down the cartel will sometimes improve society as a whole.

Escape from the Prisoner's Dilemma

With the passing of time, a range of strategies have been created to resolve the dilemmas of inmates to conquer human incentives for the greater good. Second, the natural universe is replicated more than once in most economic and other human experiences. Typically, the dilemma of a real prisoner is played once or otherwise classed as a dilemma for an iterated prisoner. The players may determine tactics to reward collaboration or to punish defection over time, in the iterated prisoner's dilemma. We can also willingly switch from a once prisoner's dilemma to a recurring prisoner's dilemma, by communicating regularly with the same individuals.

Second, structural mechanisms have been established to shift the incentives confronting particular politicians. Collective initiative to encourage cooperative initiative by prestige, policy, democracy or other collective decision-making, and clear institutional retribution for defeats is translating certain prisoners' dilemmas into the mutually more advantageous effects of collaboration.

Last but not least, certain individuals and communities have established psychological and behavioral biases over time, such as stronger belief in each other, long-term prospective orientation of recurring relationships, and attitudes to positive reciprocity in cooperative behavior, or negative reciprocity. In a social background, or community selection of different competitive firms, these patterns may develop by a certain natural selection. They actually lead groups of people to choose "irrationally" outcomes, which actually benefit them all.

Together they all contribute to overcome the multiple prisoner dilemmas we may otherwise have to confront.

Finite and infinite repeated games

Repeated games may be split overall into two, finite and eternal groups, based on the duration of the session. Finite games are those in which both players know that a certain number of rounds (and finite) have been played and that after many rounds the game ends for sure. In general, the reverse inference of finite games can be solved.

Infinite games are the ones in which the game is played indefinitely. A game with an infinite number of rounds is also equivalent to a game in which players are unaware of the number of rounds in which the game is played. Infinite games (or games which are played many times unknown) cannot be resolved by reverse induction because no "last round" is possible to begin the reverse induction from. Even if the play is identical in each round, it can generally lead to very different outcomes (equilibriums) and very different optimum strategies to repeat that game a few or more times.

Limited-average versus future-discounted reward

Think of an entity engaging in loops with a world. The agent is compensated for its success in each contact period. We compare Cycle 1 to M (average value) with Cycle K to Infinity (discounted value) for the future discounted reward V. In essence, we accept discount (non-geometric) sequences and random (non-MDP) premium sequences. We display U for m->infinity and V for k->infinity asymptotically equals, as long as all limits remain. Furthermore, the presence of the limits of U indicates that the limits of V arise if the productive horizon rises linearly with k or faster. In reverse, the existence of the limit of V means that the effectiveness of the horizon grows linearly or slowly with k or slower.

Folk theorems

Folk theorems define an excess of Nash balance payoff profiles in repeated games as a class of theorems in games. The initial Folk Theorem applied to the payoffs of the Nash balance in an infinitely repeated game. This result is called the Folk Hypothesis since, while no-one had published it, it was commonly recognized to game theorists in the 1950s. Friedman’s theorem of 1971 concerns the reward of the infinitely replicated, definite sub game-perfect Nash balance (SPE) by utilizing a more efficient definition of balance, and thus reinforces the initial Folk Theorem.

The folk theorem implies that, when the players are careful enough and far-sighted, then repetitive experiences will contribute to an average SPE balance payoff of almost any sort. All cooperating parties are not Nash equilibrium in the one-shot Dilemma of Prisoner. The only Nash balance is that both players are flawed, and often have a profile of the minmax. One folk theorem states that, as long as players are careful enough in the constantly played version of the game, there is a healthy Nash harmony such that both players work together on the road towards peace. But if only a known amount of times is replicated in a given game, it can be calculated that both players will execute the one-shoot Nash balance in each cycle, which ensures that each time they will defect.