# Payoff Information and Self-Confirming Equilibrium

## Abstract

In a self-confirming equilibrium, each player correctly forecasts the actions that opponents will take along the equilibrium path, but may be mistaken about the way that opponents would respond to deviations. Intuitively, this equilibrium concept models the possible steady states of a learning process in which, each time the game is played, players observe only the actions played by their opponents (as opposed to the complete specification of the opponents' strategies) so that they need never receive evidence that their forecasts of off-path play are incorrect. 3 Because self- confirming equilibrium (henceforth "SCE") allows beliefs about off-path play to be completely arbitrary, it (like Nash equilibrium) corresponds to a situation in which players have no prior information about the payoff fimctions of their opponents.4 This may be a good approximation of some real-world situations; it is also the obvious way to model play in game theory experiments in which subjects are given no itiormation about opponents' payoffs. In other cases, both in the real world and in the laboratory, it seems plausible that players do have some prior information about their opponents' payoffs. The goal of this paper is to develop a more restrictive version of SCE that incorporates the effects of such prior information. In carrying out this program, a key issue is what sort of prior itiormation about payoffs should be considered. It is well known that predictions based on common certainty of payoffs are not robust to even a small amount of uncertainty. Following Fudenberg, Kreps and Levine (1987), we are interested in the strongest possible assumption that is robust to payoff uncertainty. Past work suggests that this assumption should be that payoffs are almost common certainty in the sense of Monderer and Samet (1989).5 Therefore we start by developing a preliminary concept -- rationalizability at reachable nodes -- that is robust and incorporates almost common certainty of th(This abstract was borrowed from another version of this item.)

## Download Info

If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are**not**on the IDEAS site. Please be patient as the files may be large.

## Bibliographic Info

Paper provided by ESRC Centre on Economics Learning and Social Evolution in its series ELSE working papers with number 040.

**Length:**

**Date of creation:**

**Date of revision:**

**Handle:**RePEc:els:esrcls:040

**Contact details of provider:**

Postal: Gower Street, London WC1E 6BT

Email:

Web page: http://else.econ.ucl.ac.uk/

More information through EDIRC

## Related research

**Keywords:**ract: In a self-confirming equilibrium; each player correctly forecasts the actions that opponents will take along the equilibrium path; but may be mistaken about the way that opponents would respond to deviations. Intuitively; this equilibrium concept models the possible steady states of a learning process in which; each time the game is played; players observe only the actions played by their opponents (as opposed to the complete specification of the opponents' strategies) so that they need never receive evidence that their forecasts of off-path play are incorrect. 3 Because self- confirming equilibrium (henceforth "SCE") allows beliefs about off-path play to be completely arbitrary; it (like Nash equilibrium) corresponds to a situation in which players have no prior information about the payoff fimctions of their opponents.4 This may be a good approximation of some real-world situations; it is also the obvious way to model play in game theory experiments in which subjects are given no itiormation about opponents' payoffs. In other cases; both in the real world and in the laboratory; it seems plausible that players do have some prior information about their opponents' payoffs. The goal of this paper is to develop a more restrictive version of SCE that incorporates the effects of such prior information. In carrying out this program; a key issue is what sort of prior itiormation about payoffs should be considered. It is well known that predictions based on common certainty of payoffs are not robust to even a small amount of uncertainty. Following Fudenberg; Kreps and Levine (1987); we are interested in the strongest possible assumption that is robust to payoff uncertainty. Past work suggests that this assumption should be that payoffs are almost common certainty in the sense of Monderer and Samet (1989).5 Therefore we start by developing a preliminary concept -- rationalizability at reachable nodes -- that is robust and incorporates almost common certainty of the payoffs. In particular; we suppose that players believe that their opponents' actions will maximize their presumed payoff fbnctions so long as the opponents have not been observed to deviate from anticipated play. However; players do not use the prior payoff iniiormation to restrict their beliefs about the play of opponents who have already been observed to deviate from expected play. Intuitively; this corresponds to players supposing that such deviations are signals that the deviator's payoff finction is different than had been expected. (This assumes; in addition to almost common certainty of the payoffs; that the payoffs are determined independently; so that the signal refers only to the deviator's payoffs. We discuss the issue of independence in section 6.) We verify in section 3 that this intuition is correct and that this concept is robust; in section 4 we show that it does correspond to assuming almost common certainty of payoffs and independence. To capture the idea of SCE; namely that play corresponds to the steady state of a learning process in which the path is observed each time the game is played; we add the assumption that the path of play is public information. This is in the spirit of but stronger than; the assumption underlying self- confirming equilibrium; which is that each player knows the path of play. For simplicity; we also impose the assumption that players' beliefs concerning their opponents' play correspond to independent randomizations. (As mentioned earlier; independence is discussed fi.u-ther in section 6.) Combining these assumptions lead to rationalizable self-confirming equilibrium; or "RSCE." If we think of equilibrium as describing the steady state of a random- matching process of the sort used in most game theory experiments; then there are many agents; who are allocated into groups; one for each player role in the game. The observations of different agents in the same player role may differ depending on the actions they take. If itiormation about the aggregate distribution of outcomes is not available to the subjects; as it is not in most experiments; and players observe only the outcomes in their own matches; then the appropriate notion of SCE is Fudenberg and Levine's (1993a) notion of a heterogeneous SCE.G The corresponding notion with prior payoff information is a heterogeneous RSCE. Perhaps the best motivation and illustration of our ideas is a pair of experiments by Prasnikar and Roth [1992] on the "best-shot" game. in which two players sequentially decide how much to contribute to a public good. The backwards-induction solution to this game is for player 1 to contribute nothing and player 2 to contribute 4; this is also the only RSCE. There is also an imperfect Nash equilibrium in which player 1 contributes 4 and player 2 contributes nothing. Prasnikar and Roth ran two treatments of this game. In the first one; players were informed of the fimction determining opponents' monetary payoffs. Here; by the last few rounds of the experiment the first movers had stopped contributing; which is the prediction made by RSCE. In the second treatment; subjects were not given any irdlormation about the payoffs of their opponents. In this treatment even in the later rounds of the experiment many first movers contributed to the public good. This is not consistent with RSCE; but it is consistent with an (approximate; heterogeneous) SCE (Fudenberg and Levine [1996] 7 ). Thus these experiments provide evidence that information about other players' payoffs makes a difference; and that this difference corresponds to the distinction between SCE and RSCE.8 Papers by Rubinstein and Wolinksy [1994] and Greenberg [1994]; like ours; are based on the idea that players form their forecasts of opponents play using prior information both about the opponents' payoffs and about the realized outcomes when the game is played. Both these papers; unlike ours; consider common certainty of rationality; while our desire for robust predictions leads us to consider almost common certainty instead. Like us; Greenberg works in an extensive-form context; with players observing terminal nodes but not intended off-path play; Rubinstein and Wolinksy consider strategic-form games and general "signal fictions". We should make clear from the outset that; although this paper is motivated by the learning-theoretic approach to equilibrium in games; we do not here provide an explicit learning-theoretic foundation for our concepts. We are confident that such foundations can be constructed by; for example; incorporating restrictions on the priors into the steady-state learning model of Fudenberg and Levine [1993 b]; but we have not checked the details.;

**Other versions of this item:**

- Dekel, Eddie & Fudenberg, Drew & Levine, David K., 1999.
"
**Payoff Information and Self-Confirming Equilibrium**," Journal of Economic Theory, Elsevier, vol. 89(2), pages 165-185, December.

- Eddie Dekel & Drew Fudenberg & David K. Levine, 1996.
"
**Payoff Information and Self-Confirming Equilibrium**," Harvard Institute of Economic Research Working Papers 1774, Harvard - Institute of Economic Research. - Eddie Dekel & Drew Fudenberg & David K. Levine, 1999.
"
**Payoff Information and Self-Confirming Equilibrium**," Levine's Working Paper Archive 172, David K. Levine. - Eddie Dekel & Drew Fudenberg & David K. Levine, .
"
**Payoff Information and Self-Confirming Equilibrium**," ELSE working papers 032, ESRC Centre on Economics Learning and Social Evolution. - Levine, David & Dekel, Eddie & Fudenberg, Drew, 1999.
"
**Payoff Information and Self-Confirming Equilibrium**," Scholarly Articles 3200614, Harvard University Department of Economics. - Dekel, E. & Fudenberg, D. & Levine, D.K., 1999.
"
**Payoff information and Self-Confirming Equilibrium**," Papers 9-99, Tel Aviv.

**Find related papers by JEL classification:**

**C7**- Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory

## References

**References listed on IDEAS**

Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:

- Kreps, David M & Wilson, Robert, 1982.
"
**Sequential Equilibria**," Econometrica, Econometric Society, vol. 50(4), pages 863-94, July.

- David Kreps & Robert Wilson, 1998.
"
**Sequential Equilibria**," Levine's Working Paper Archive 237, David K. Levine. - David M Kreps & Robert Wilson, 2003.
"
**Sequential Equilibria**," Levine's Working Paper Archive 618897000000000813, David K. Levine.

- David Kreps & Robert Wilson, 1998.
"
- Borgers Tilman, 1994.
"
**Weak Dominance and Approximate Common Knowledge**," Journal of Economic Theory, Elsevier, vol. 64(1), pages 265-276, October.

- T. Börgers, 2010.
"
**Weak Dominance and Approximate Common Knowledge**," Levine's Working Paper Archive 378, David K. Levine.

- T. Börgers, 2010.
"
- Drew Fudenberg & David M. Kreps & David K. Levine, 1986.
"
**On the Robustness of Equilibrium Refinements**," UCLA Economics Working Papers 398, UCLA Department of Economics.

- Fudenberg, Drew & Kreps, David M. & Levine, David K., 1988.
"
**On the robustness of equilibrium refinements**," Journal of Economic Theory, Elsevier, vol. 44(2), pages 354-380, April.

- Levine, David & Kreps, David & Fudenberg, Drew, 1988.
"
**On the Robustness of Equilibrium Refinements**," Scholarly Articles 3350444, Harvard University Department of Economics. - Drew Fudenberg & David Kreps & David K. Levine, 1988.
"
**On the Robustness of Equilibrium Refinements**," Levine's Working Paper Archive 227, David K. Levine.

- Fudenberg, Drew & Kreps, David M. & Levine, David K., 1988.
"
- Werlang, Sérgio Ribeiro da Costa & Chin-Chiu Tan, Tommy, 1987.
"
**The Bayesian Foundations of Solution Concepts of Games**," Economics Working Papers (Ensaios Economicos da EPGE) 111, FGV/EPGE Escola Brasileira de Economia e Finanças, Getulio Vargas Foundation (Brazil).

- Tan, Tommy Chin-Chiu & da Costa Werlang, Sergio Ribeiro, 1988.
"
**The Bayesian foundations of solution concepts of games**," Journal of Economic Theory, Elsevier, vol. 45(2), pages 370-391, August.

- Tan, Tommy Chin-Chiu & da Costa Werlang, Sergio Ribeiro, 1988.
"
- D. Pearce, 2010.
"
**Rationalizable Strategic Behavior and the Problem of Perfection**," Levine's Working Paper Archive 523, David K. Levine. - Drew Fudenberg & Jean Tirole, 1991.
"
**Game Theory**," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262061414, December. - Pearce, David G, 1984.
"
**Rationalizable Strategic Behavior and the Problem of Perfection**," Econometrica, Econometric Society, vol. 52(4), pages 1029-50, July. - Fudenberg, D. & Levine, D.K., 1991.
"
**Self-Confirming Equilibrium**," Working papers 581, Massachusetts Institute of Technology (MIT), Department of Economics.

- Fudenberg, Drew & Levine, David K, 1993.
"
**Self-Confirming Equilibrium**," Econometrica, Econometric Society, vol. 61(3), pages 523-45, May.

- Drew Fudenberg & David K. Levine, 1993.
"
**Self-Confirming Equilibrium**," Levine's Working Paper Archive 2147, David K. Levine.

- Fudenberg, Drew & Levine, David K, 1993.
"
- Fudenberg, Drew & Levine, David K, 1993.
"
**Steady State Learning and Nash Equilibrium**," Econometrica, Econometric Society, vol. 61(3), pages 547-73, May.

- Drew Fudenberg & David K. Levine, 1993.
"
**Steady State Learning and Nash Equilibrium**," Levine's Working Paper Archive 373, David K. Levine.

- Drew Fudenberg & David K. Levine, 1993.
"
- Bernheim, B Douglas, 1984.
"
**Rationalizable Strategic Behavior**," Econometrica, Econometric Society, vol. 52(4), pages 1007-28, July.

- D. B. Bernheim, 2010.
"
**Rationalizable Strategic Behavior**," Levine's Working Paper Archive 661465000000000381, David K. Levine. - D. B. Bernheim, 2010.
"
**Rationalizable Strategic Behavior**," Levine's Working Paper Archive 514, David K. Levine.

- D. B. Bernheim, 2010.
"
- Gul, Faruk, 1996.
"
**Rationality and Coherent Theories of Strategic Behavior**," Journal of Economic Theory, Elsevier, vol. 70(1), pages 1-31, July. - Basu, Kaushik, 1988.
"
**Strategic irrationality in extensive games**," Mathematical Social Sciences, Elsevier, vol. 15(3), pages 247-260, June.

- Kaushik Basu, 2010.
"
**Strategic Irrationality in Extensive Games**," Levine's Working Paper Archive 375, David K. Levine.

- Kaushik Basu, 2010.
"
- Lawrence E. Blume & William R. Zame, 1993.
"
**The Algebraic Geometry of Perfect and Sequential Equilibrium**," Game Theory and Information 9309001, EconWPA.

- Blume, Lawrence E & Zame, William R, 1994.
"
**The Algebraic Geometry of Perfect and Sequential Equilibrium**," Econometrica, Econometric Society, vol. 62(4), pages 783-94, July.

- Blume, Lawrence E & Zame, William R, 1994.
"
- A. Rubinstein & A. Wolinsky, 2010.
"
**Rationalizable Conjectural Equilibrium: Between Nash and Rationalizability**," Levine's Working Paper Archive 369, David K. Levine.

- Rubinstein Ariel & Wolinsky Asher, 1994.
"
**Rationalizable Conjectural Equilibrium: Between Nash and Rationalizability**," Games and Economic Behavior, Elsevier, vol. 6(2), pages 299-311, March.

- Ariel Rubinstein & Asher Wolinsky, 1991.
"
**Rationalizable Conjectural Equilibrium: Between Nash and Rationalizability**," Discussion Papers 933, Northwestern University, Center for Mathematical Studies in Economics and Management Science.

- Rubinstein Ariel & Wolinsky Asher, 1994.
"
- Drew Fudenberg & Eddie Dekel, 1987.
"
**Rational Behavior with Payoff Uncertainty**," Working papers 471, Massachusetts Institute of Technology (MIT), Department of Economics.

- Dekel, Eddie & Fudenberg, Drew, 1990.
"
**Rational behavior with payoff uncertainty**," Journal of Economic Theory, Elsevier, vol. 52(2), pages 243-267, December.

- E. Dekel & D. Fudenberg, 2010.
"
**Rational Behavior with Payoff Uncertainty**," Levine's Working Paper Archive 379, David K. Levine.

- Dekel, Eddie & Fudenberg, Drew, 1990.
"
- Costa-Gomes, Miguel & Crawford, Vincent P. & Broseta, Bruno, 1998.
"
**Cognition and Behavior in Normal-Form Games: An Experimental Study**," University of California at San Diego, Economics Working Paper Series qt1vn4h7x5, Department of Economics, UC San Diego.

- Costa-Gomes, Miguel & Crawford, Vincent P & Broseta, Bruno, 2001.
"
**Cognition and Behavior in Normal-Form Games: An Experimental Study**," Econometrica, Econometric Society, vol. 69(5), pages 1193-1235, September.

- Miguel Costa-Gomes & Vincent P. Crawford & Bruno Broseta, .
"
**Cognition and Behavior in Normal-Form Games:An Experimental Study**," Discussion Papers 00/45, Department of Economics, University of York. - Broseta, Bruno & Costa-Gomes, Miguel & Crawford, Vincent P., 2000.
"
**Cognition and Behavior in Normal-Form Games: An Experimental Study**," University of California at San Diego, Economics Working Paper Series qt0fp8278k, Department of Economics, UC San Diego.

- Costa-Gomes, Miguel & Crawford, Vincent P & Broseta, Bruno, 2001.
"
- D. Fudenberg & D. M. Kreps, 2010.
"
**Learning in Extensive Games, I: Self-Confirming Equilibrium**," Levine's Working Paper Archive 382, David K. Levine. - Monderer, Dov & Samet, Dov, 1989.
"
**Approximating common knowledge with common beliefs**," Games and Economic Behavior, Elsevier, vol. 1(2), pages 170-190, June. - Reny Philip J., 1993.
"
**Common Belief and the Theory of Games with Perfect Information**," Journal of Economic Theory, Elsevier, vol. 59(2), pages 257-274, April.

- P. Reny, 2010.
"
**Common Belief and the Theory of Games with Perfect Information**," Levine's Working Paper Archive 386, David K. Levine.

- P. Reny, 2010.
"
- Battigalli, Pierpaolo, 2003.
"
**Rationalizability in infinite, dynamic games with incomplete information**," Research in Economics, Elsevier, vol. 57(1), pages 1-38, March. - Ben-Porath, Elchanan, 1997.
"
**Rationality, Nash Equilibrium and Backwards Induction in Perfect-Information Games**," Review of Economic Studies, Wiley Blackwell, vol. 64(1), pages 23-46, January. - Philip J. Reny, 1992.
"
**Rationality in Extensive-Form Games**," Journal of Economic Perspectives, American Economic Association, vol. 6(4), pages 103-118, Fall.

**Full references**(including those not matched with items on IDEAS)

## Citations

Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.**Cited by:**

**This item has more than 25 citations. To prevent cluttering this page, these citations are listed on a separate page.**

## Lists

This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.## Statistics

Access and download statistics## Corrections

When requesting a correction, please mention this item's handle: RePEc:els:esrcls:040. See general information about how to correct material in RePEc.For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (s. malkani) The email address of this maintainer does not seem to be valid anymore. Please ask s. malkani to update the entry or send us the correct address.

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

If references are entirely missing, you can add them using this form.

If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.

Please note that corrections may take a couple of weeks to filter through the various RePEc services.