Talk:Centipede game

Game Tree is not Correct

The payoffs indicated in the game tree are not correct. In the last round, the payoff to defection should be 4 for player 2, not zero - as currently drawn, SPNE would have player 2 cooperate in this round. A version with correct payoffs can be found e.g. on the Italian Wikipedia - http://it.wikipedia.org/wiki/File:Centipede_game.png - perhaps it is possible to just use that drawing instead? — Preceding unsigned comment added by 185.11.153.254 (talk) 13:28, 1 April 2014 (UTC)

Agreed, this looks wrong. I've swapped the image as suggested. --McGeddon (talk) 21:12, 8 December 2016 (UTC)

After deletion

After deletion of copyvio material, redirect to Centipede (video game). SWAdair | Talk 11:11, 15 Nov 2004 (UTC)

Game theory is referring to a different "centipede game", so maybe that's not such a good idea. --rbrwr^± 22:24, 16 Nov 2004 (UTC)

Ah, I see what you mean. SWAdair | Talk 05:25, 17 Nov 2004 (UTC)

Equilibrium

While the unique solution for a game of certain length is obviously to defect on the first go, this is only a solution for a game of certain length - a game of uncertain length works a bit differently. Also, technically, if it works the way it states in this article (pass one coin across the table; if your opponent defects they get two coins and you still get one) then the unique equilibrium is to pass it back and forth infinitely, as you're never losing anything by passing. Only if you get more by defecting at any given point (rather than equal) is it actually at the stated equilibrium. Titanium Dragon 08:47, 11 January 2007 (UTC)

Contradictory

The introduction states that "The payoffs are arranged so that if one passes the pot to one's opponent and the opponent takes the pot, one receives slightly less than if one had taken the pot", yet the example in the explanation of the rules starts with a pot of 1 and pot of 0. This would iterate to 2 and 1 on the first pass, 3 and 2 on the second pass, and so on - player 1 would never lose money via passing. I reworded the explanation to begin with pots of 2 and 0 (hence, the first player would receive 1 less token than he passed if the second player cashes out), but the graph may be inaccurate. I believe the graph to be beyond my meager means to possibly interpret and rule on. Somebody might want to verify that the graph is still accurate / inaccurate, and fix if necessary. --Action Jackson IV 12:24, 5 February 2007 (UTC)

Thanks for fixing it. It looks good now. --best, kevin [kzollman][talk] 20:38, 5 February 2007 (UTC)

Why all the talk of Subgame perfect equilibriums

The Subgame perfect equilibrium says "However, backward induction cannot be applied to games of imperfect or incomplete information because this entails cutting through non-singleton information sets."

Surely you don't know what the other persons decisions are so you have incomplete knowledge so the Subgame perfect equilibrium solution is a pretty silly solution to take. It is an obvious policy if you want a no regrets policy at the expense of loosing all hope of a possibly much more lucrative solution.

The obvious starting point is that the possibility of loosing 1 out of 2 a 50% loss could quite easily be compensated for by a fairly low possibility of a solution that gives around say 50 rather than 2. If the game only has 5 rounds then ending the game at the first opportunity looks more sensible. If there are 1000 rounds who would choose to default at the first opportunity? Anyway if the game has many rounds and both players think in these terms then it seems quite reasonable to say that the further on the game gets the more likely the players should be to end the game rather than continue because the possible benefit of continuing reduces while the potential loss of 1 at each node stays the same. For this game, I don't think there is any getting away from assessing the probability of the other players decisions.

Once the first player has passed both piles, this shows an intent to co-operate at least for a while. This should help get the co-operation going and may affect the probability assessments. If you have to decide your policy in advance this should be taken into consideration.

Anyway all this talk of the Subgame perfect equilibrium seems a bit silly if it is clear that it does not apply. crandles 14:05, 6 August 2007 (UTC)

The page says "However, analysis shows a different outcome; namely that the best decision for the first player is to pocket the pile of two coins on the first round, as explained below:". That sounds like pretty obviously wrong analysis to me. crandles 14:19, 6 August 2007 (UTC)

Incidentally, how much would you pay to get a seat at this game assuming the other player has also paid a similar amount and therefore faces the same incentives/dilemas as you do? (assume 100 rounds) crandles 14:19, 6 August 2007 (UTC)

The McKelvey and Palfrey (1992) paper covers this in depth, and is mentioned in the article. I don't think there is any reason we should lessen the emphasis of the SPE (or the equilibrium analysis), in fact your comment may be seen as a sort of feature rather than a bug. The model of the centipede game shows that in certain situations, people may only rarely (6% of the time according to McKelvey and Palfrey) play SPEs. Originally, you recall, the centipede game was proposed in Rosenthal (1981) as a model to explain the chain-store paradox (Selton, The Chain Store Paradox, Theory and Decision 9 (1978) 127-159), which may be a more sensible way to see the application of the model and solution. I think that the efforts to resolve the problem more social cases are to say that this game as a model of certain real situations and SPE as a model of rules for making decisions are problematic. Specifically, if bounded rationality is used rather than "rationality" (in as much as an SPE is "rational"), then perhaps the problem is averted.

I don't know the answer to your second question off hand. Smmurphy^(Talk) 16:04, 6 August 2007 (UTC)

OK it is a feature not a bug. So should the article be rewritten to say the game shows an example of a case where the SPE gives a wrong answer and explaination(s) of why the SPE fails rather than the current article which seems to suggest the SPE analysis is correct and people playing the game differently are irrational/altruistic? If there were more than 50 rounds to go I would certainly try to continue the game. The reason for this is not altruism, I am trying to get a bigger payout. I don't believe it is irrational because I know there is some information that I cannot deduce and trying to deduce it with silly logic that pretends it can work out everything is not sensible. OK I know that saying a paradox is just silly may not explain the paradox. It is the explanation of the paradox that is important and this seems lacking in the current article. crandles 18:29, 6 August 2007 (UTC)

If you don't know the answer, are you willing to suggest whether it is over 5? If not, and you are aware of the 6% figure then I suggest this implies you are exceedingly risk averse. crandles 18:34, 6 August 2007 (UTC)

SPE is only a the "correct" thing to do if it is common knowledge that both players will engage in backwards induction. So, if you know that the other player will play the SPE, defecting on the first round will maximize your utility. However, if you assign some positive probability to the other player playing across on the next round, it might be right to play across. Your first point about the first person passing, is an important issue in the discussion of subgame perfection, that is: suppose someone takes an action which is not a component of an SPE. What is the right thing to infer from that (a) that they won't play the SPE in the future or (b) that they made a mistake and are likely to continue playing an SPE? There is a lot of discussion about this issue in the philosophical literature, but is well beyond the scope of this article.

That aside, you say "So should the article be rewritten to say the game shows an example of a case where the SPE gives a wrong answer and explaination(s) of why the SPE fails rather than the current article which seems to suggest the SPE analysis is correct and people playing the game differently are irrational/altruistic?" I think it already does that. The lead mentions this explicitly as does the first sentence of the "explanations" section. Your explanation is mentioned in the second sentence of that section and the second and third paragraphs ofter alternative explanations. If you would like to expand those sections, please do so (although be cautions of original research). But personally, I don't think the article fails to address the issues your raising. --best, kevin [kzollman][talk] 21:59, 6 August 2007 (UTC)

I have tried to explain the rules more clearly. You may have felt it was adequately explained but I felt it needed to state more specificically what it is about and I have had a go at this. The Explanations of empirical results section that seems to require altruism or error to explain the reality of the game still makes me cringe but if it is in the litrature then I cannot really edit it without becoming a lot more familiar with the litrature or saying something that ought to be deleted as original research. crandles 23:05, 6 August 2007 (UTC)

I reworded the change in the lead to be more precise. To say that subgame perfection is "imperfect" is a bit overly broad. I removed the two added paragraphs to the rules section. It seemed to me to make the article more disorganized (by introducing something which is then repeated in the later section) and also presumes that the reader would expect the game to go the full 100 round (which almost never happens). --best, kevin [kzollman][talk] 01:13, 7 August 2007 (UTC)

Since you part reverted my changes to the rules section, I have now more fully reverted. As you can see it previously suggested 100 rounds rather than close to 100 rounds so the problem you complain about was there before I made edits and I alleviated the problem. So you are agreeing with me but reverting me!? I also thought my order made more sense. I have now attempted a different set of edits to try to alleviate the problem again. crandles 08:05, 7 August 2007 (UTC)

There are three distinct issues that you are conflating which is leading to a misunderstanding. Here are the three questions: (1) As a matter of mathematical fact what are the Nash equilibria and Subgame perfect equilibria of this game. This issue is not a matter of dispute, all the NE and the unique SPE involve the first player defecting on the first round. (2) Do the NE and SPE correspond with how people actually play the game? Studies seem to show that they don't. (3) Do the NE and SPE provide a recommendation for proper play, i.e. should people play the NE or SPE? This is the normative part of game theory. This one is more difficult, but perhaps worth discussing. Importantly, there are conditions under which the game theoretic solution is the right thing to do (while there are also conditions under which it is the wrong thing to do).

The reason I point these three out is that calling game theory solutions "wrong" confuses these three issues. Your recent edits are imprecise in exactly this way, and so as a result are misleading. This is why I removed/edited them to be more clear. --best, kevin [kzollman][talk] 18:43, 7 August 2007 (UTC)

Ok I will accept all of those reverts except for the 'dictates' word. It doesn't do a good job of dictating if only 6% follow that and I don't see how a simple change to 'indicate' could be conflating.crandles 19:32, 7 August 2007 (UTC)

Sounds good. --best, kevin [kzollman][talk] 19:34, 7 August 2007 (UTC)

Name of article

Should this article be named 'Implications of the centipede game for game theory'? If not I suggest it could be argued that there should be much more about probability theory attempts at a solution and how the game should be played rather than about silly wrong game theory solutions? crandles 13:34, 7 August 2007 (UTC)

The title is fine, and I agree that other solutions than SPE could be discussed more. But be careful that you are discussing the attempts of others (by citing whose attempt, etc) so that you avoid original research. As for probability theory solutions rather than GT ones, I think that all solutions are GT solutions, just they use different sets of rules and different methodologies (mixed strategies, bounded rationality, or whatever). I'll have time to add something along these lines myself tomorrow, probably. Best, Smmurphy^(Talk) 11:31, 8 August 2007 (UTC)

Explanation for empirical results

The definition of the problem assumes that gains for the other player have no value for the first player.

This is not true in practice. It would be very difficult to separate these components of social interactions from the game. It is not altruism so much as reinforcement of social harmony, which has direct if hard-to-describe-benefits to the player. Even if you had a situation with an anonymous opponent over e.g. the internet: why are people playing the game at all? If it is to get a high 'score' or large point gain, then the value of playing is directly tied into whether the first player feels they can get a good score. A low score is worthless in a competitive environment, and so strategies which result in low scores are automatically discarded. If this means always losing the pot to an opponent who immediately defects, then the player will feel they have no choice but to stop playing the game, it is just one option in the the 'real world' of more complex choices after all. —Preceding unsigned comment added by 96.26.192.159 (talk) 22:07, 11 May 2008 (UTC) <== oops. =p

I don't get why anyone thinks that altruism has anything to do with it. It's self evident that the reason for not defecting on the first round is that the amount you would get is at the seconds smallest on the first round. You're being selfish, thinking ahead to the next round. You also assume the other person is thinking the same way. When you become unsure of this is when you finally take the pot. The fact that only only get slightly less than you would otherwise makes this risk not as great, but it does mount each time.

Sometimes I think game theorists miss the forest for the trees. — trlkly 06:54, 15 April 2012 (UTC)

Auctions?

Perhaps there could be some mention of how in an auction where the player has malicious will against the other bidder, the same sort of situation arises. By waiting longer, they increase the cost of to their opponed, but by waiting longer, they end up with the inflated cost instead. In this case, the "cooperation" of the centepede game is equivilent to non-cooperation in this game.

71.161.146.225 (talk) 23:17, 13 September 2008 (UTC)

Slowly increasing pot

The introduction speaks of a slowly increasing pot when the description presents an exponentially increasing pot. --PIerre.Lescanne (talk) 08:28, 27 August 2015 (UTC)

I removed "slowly".--PIerre.Lescanne (talk) 15:37, 5 October 2015 (UTC)