Talk:Folk theorem (game theory)

Why is the Folk Theorem of Repeated Prisoner's Dilemma so important

Why is the Folk Theorem of Repeated Prisoner's Dilemma so important ?

There are many analogies between Repeated Prisoner's Dilemma (with an unknown end-round) and issues of competition, cooperation and coordination. Repeated/Iterated Prisoner's dilemma is widely used as a model in economics, business, psychology, sociology political science, and other social and information sciences. (see prisoner's dilemma / tragedy of the commons / market failure / Leviathan / public goods ). It is also widely used to describe cooperation and/or competition within and/or between species in an evolutionary setting.

Prisoner's dilemma games are characterized by each player choosing to Defect (D) or Cooperate (C). The highest payoff is for Defecting against cooperating opponents (Exploit). The lowest payoff is for cooperating against defecting opponents (Sucker). Both players defecting (mutual defection (MD)) results in lower payoffs to each player than both player cooperating (mutual cooperation (MC)). So, while going from the stable equilibrium of MD to MC would be a Pareto improvement, in single-shot Prisoner's dilemma rational economic players fail to achieve the efficiency of mutual cooperation.

Fundamentally, single-game Prisoner's dilemma demonstrates where the "invisible hand" of competition can fumble - creating an outcome that is inefficient for all players. (Note: single-game Prisoner's dilemma is fundamentally different than mixed-motive games and pure coordination games - which describe yet other situations where a pure competitive approach may produce distinctly inefficient results - see coordination games regarding the importance of focal points and/or of explicit coordination / standards).

Repeated Prisoner's dilemma (with an unknown end-round) allows for the possibility of VARYING DEGREES of cooperation and defection. The FOLK THEOREM demonstrates this key point. ANY equilibrium that pays each player an average payoff of at least the mutual defection payoff (+ epsilon) is supportable. (Cf. Folk Theorem entry regarding "Grim strategy" approach). NOTE : this does NOT imply that both players will necessarily achieve the most efficient average payoff (usually defined as the MC payoff - since typically PD games are defined so that 2*MC >= Sucker + Exploit). In fact, it states - far more broadly and far more interestingly - that ANY equilibrium in a very broad range MAY be supported. So the Folk Theorem for Repeated Prisoner's Dilemma (with unknown end round) can be used to model unequal and even ongoing expoitative relationships - preditor-prey, symbiotic or parasitic, bully-victim, noble-serf, etc. For example, if a given player (noble) is precommitted to (see entry on precommitment, and on Thomas Schelling) a grim strategy that will require the other player (serf) to allow him/herself to be exploited every k rounds, then as long as the other player (serf) gets an average payoff of AT LEAST the MD payoff + epsilon, it is in his/her interest NOT to violate the pattern of play, since triggering the GRIM outcome will result in the MD average payoff (which is dominated). This means that the disadvantaged player may have an economic interest in maintaining his/her unequal and exploited position rather than face mutual defection. (See again the advantages of precommittment in a game-theoretic or negotiation context).

Ironically, the FOLK THEOREM implies that the meta-game of Repeated Prisoner's dilemma (with an unknown end-round) is actually a mixed-motive coordination game. (See entries on coordination games and Thomas Schelling). This implies that all the tactics that may be of use in mixed motive games may apply to Repeated Prisoner's dilemma games (with an unknown end-round).

Practically, why does that matter ? Repeated Prisoner's dilemma games (with an unknown end-round) model a very broad range of situations where players - whether individuals, businesses, countries or organisms - can cooperate or fight.

(True single-game prisoner's dilemma situations are probably rare - most interactions can at least cause reputation effects, repeated games on the other hand are quite common). The FOLK THEOREM shows us - in game theory - what we observe in practical reality - that expectations and beliefs matter, that initial positions matter, that precommitment matters. It can explain the wide range of equilibria that we observe in people's interactions across and within cultures, across and within organizations, and across and within species. It provides a much richer model for economic behavior than older approaches.

                                                                (Holt, G.)

Welcome to wikipedia and thank you for your addition. If you would like this to be included in the article itself, you are welcome to add it. Although, it would be useful if it provided some sources. Not that I distrust anything said here, quite the contrary. However, wikipedia prohibits the posting of original research and so I would like to be sure this does not include any of that. Again thank you for your interest. --best, kevin [kzollman][talk] 22:40, 19 January 2007 (UTC)[reply]

While players may have incentive to stay in disadvantageous positions which are still better than hardcore mutual defection, the game being a coordination game implies that there are multiple Nash equilibria, and it is perfectly rational to attempt to shift from one equilibrium to another, even if it leads to miscoordination with awful payoffs. Worth remembering that being rational and being wise are not the same thing. Wat 20 02:55, 25 June 2014 (UTC)[reply]

And thanks A LOT for classifying the Folk Theorem as a coordination game. Explains very well why simply trying to enforce something backfires a lot more often than most "iron fist" theories predict. Enforcement is not enough. All parts must agree to the same equilibrium first. Otherwise, miscoordination arises. Wat 20 02:55, 25 June 2014 (UTC)[reply]

A solution concept?

I'm a bit uncomfortable describing Folk Theorems as "solution concepts" within game theory. Perhaps I'm being pedantic, but my understanding is that Folk Theorems state that, under certain conditions, a wide variety of outcomes can be sustained as Nash equilibria in a repeated game. While Folk Theorems extend the possibilities for the solutions that might result for a particular game, I'd argue that Folk Theorems are not themselves solution concepts in the same sense as the Nash equilibrium, evolutionarily stable strategy, etc. Any thoughts? Mateoee 21:18, 8 May 2007 (UTC)[reply]

Intractability

The article ought to mention that the strategies alluded to in the article are, in general, intractable to compute.^[1] Gdr 20:53, 8 March 2008 (UTC)[reply]

^ Christian Borgs, Jennifer Chayes, Nicole Immorlica, Adam Tauman Kalai, Vahab Mirrokni, and Christos Papadimitriou (2007). "The Myth of the Folk Theorem" (PDF).{{cite web}}: CS1 maint: multiple names: authors list (link)

split off

This article as of now is just general enough to be confusing and specific enough to conflate several different concepts. Maybe splitting off soe of the ideas to separate articles, for example, Folk Theorem (Bertrand) or something could solve the problem.radek (talk) 08:14, 4 April 2010 (UTC)[reply]

Significance

In the top right box, it says the "Significance" is that Ariel Rubinstein proposed the folk theorem. Surely it has more significance, like the first comment in this discussion.

Maybe "significance" should read "Possibly robs game theory of predictive power".

This could open up a new section about disputes ... google has a lot of results on the disputatiousness of conclusions arrived at via the folk theorem. Crasshopper (talk) 23:22, 5 April 2011 (UTC)[reply]

Where's the theorem?

Article should at least state the theorem and sketch a proof that makes some kind of sense. The subgame-perfect version of the folk theorem could also be stated, and maybe more sophisticated versions with incomplete-information assumptions can be alluded to. Mct mht (talk) 09:44, 17 January 2013 (UTC)[reply]

Proof Sketch of Prefect Folk Theorem

"3. If no players deviated from phase 2, all player j ≠ i gets rewarded ε above j's minmax forever after, while player i continues receiving his minmax. (Full-dimensionality and the interior assumption is needed here.)"

This seems to be wrong. If we do this, player i would deviate from this phase, because there is no guarantee that player i is playing his single-stage best response in this phase, and he cannot be punished further because he is already receiving his minimax forever. Instead, player i need to receive v_i, some value more than his minimax, while all player j ≠ i receiving v_j+ε.

Source: http://www.virtualperfection.com/gametheory/5.3.FolkTheoremSampler.1.0.pdf (already in the references) — Preceding unsigned comment added by 167.220.232.21 (talk) 12:18, 27 April 2014 (UTC)[reply]

Folk Theorem and the Core

What if multiple players deviate at the same time? By "5. Ignore multilateral deviations." looks like the strategy doesn't take in account deviations by coalitions. Wat 20 02:28, 25 June 2014 (UTC)[reply]

Folk Theorem and Trembling hand

Is the enforceable outcome stable in the presence of noise? Grim trigger strategies are notorious for performing poorly in the presence of noise. Wat 20 02:28, 25 June 2014 (UTC)[reply]

to technical and full of jargon =

sadly , like far to many mathy wiki articles, this one is not written for the general audience, but instead is full of jargon that makes it appropriate for someone with a fair bit of math and game theory please, do better — Preceding unsigned comment added by 50.245.17.105 (talk) 14:47, 14 March 2022 (UTC)[reply]

[1] Christian Borgs, Jennifer Chayes, Nicole Immorlica, Adam Tauman Kalai, Vahab Mirrokni, and Christos Papadimitriou (2007). "The Myth of the Folk Theorem" (PDF).{{cite web}}: CS1 maint: multiple names: authors list (link)

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