## Unbeatable Imitation ∗ (2010)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Duersch10unbeatableimitation,

author = {Peter Duersch and Jörg Oechssler and Jel-classifications C and We Thank Carlos Alós-ferrer and Chen Bo and Drew Fudenberg and Er Matros and John Stachurski},

title = {Unbeatable Imitation ∗},

year = {2010}

}

### OpenURL

### Abstract

We show that for many classes of symmetric two-player games, the simple decision rule “imitate-the-best ” can hardly be beaten by any other decision rule. We provide necessary and sufficient conditions for imitation to be unbeatable and show that it can only be beaten by much in games that are of the rock-scissors-paper variety. Thus, in many interesting examples, like 2x2 games, Cournot duopoly, price competition, rent seeking, public goods games, common pool resource games, minimum effort coordination games, arms race, search, bargaining, etc., imitation cannot be beaten by much even by a very clever opponent. Keywords: Imitate-the-best, learning, symmetric games, relative payoffs, zerosum games, rock-paper-scissors, finite population ESS, potential games, quasisubmodular games, quasisupermodular games, quasiconcave games, aggregative games.

### Citations

1910 | A Course in Game Theory - Osborne, Rubinstein - 1994 |

735 | Evolutionary Game Theory - Weibull - 1995 |

374 | Supermodularity and Complementarity - Topkis - 1998 |

253 | Two person co-operative games - Nash - 1953 |

235 |
Efficient rent seeking. In
- Tullock
- 1980
(Show Context)
Citation Context ...’s probability of winning is proportional to her bid, x x+y zero if both players bid zero. The cost of bidding equals the bid. The symmetric payoff function is given by π(x, y) = x v − x (see Tullock =-=[27]-=- and Hehenkamp, Leininger, and x+y Possajennikov [11]. This game is an aggregative quasisubmodular game (see Schipper [24, Example 6] and Alós-Ferrer and Ania [1, Example 2] and π(x, y) is concave in ... |

214 | Rationalizability, learning, and equilibrium in games with strategic complementarities - Milgrom, Roberts - 1990 |

192 | Monotone comparative statics
- Milgrom, Shannon
- 1994
(Show Context)
Citation Context ...Π(x ′ , z ′′ ) > 0 ⇒ (⇐) Π(x ′′ , z ′ ) − Π(x ′ , z ′ ) > 0. (8) Quasisupermodularity (resp. quasisubmodularity) is sometimes also called the (dual) single crossing property (e.g. Milgrom and Shannon =-=[15]-=-). 10 A finite aggregative game is quasiconcave (or single-peaked) if for any x, x ′, x ′′ ∈ X with x < x ′ < x ′′ and z ∈ Z, Π(x ′ , z) ≥ min{Π(x, z), Π(x ′′ , z)}. A finite aggregative game is quasi... |

168 | Tacit Coordination Games, Strategic Uncertainty, and Coordination Failure
- Huyck, Battalio, et al.
- 1990
(Show Context)
Citation Context ...oordination) Consider the class of minimum effort games given by the symmetric payoff function π(x, y) = min{x, y} − c(x) for some cost function c(·) (see Bryant [7] and Van Huyck, Battalio, and Beil =-=[28]-=-). Corollary 3 implies that imitation is essentially unbeatable. Example 7 (Synergistic Relationship) Consider a synergistic relationship among two individuals. If both devote more effort to the relat... |

166 |
Bounded Rationality: The Adaptive Toolbox
- Gigerenzer, Selten, et al.
- 2001
(Show Context)
Citation Context ...sche 1 Introduction Psychologists and behavioral economists stress the role of simple heuristics or rules for human decision making under limited computational capabilities (see Gigerenzer and Selten =-=[10]-=-). While such heuristics lead to successful decisions in some particular tasks, they may be suboptimal in others. It is plausible that decision makers may cease to adopt heuristics that do worse than ... |

90 |
The Evolution of Walrasian Behavior
- Vega-Redondo
- 1997
(Show Context)
Citation Context ...gainst such an heuristic. The behavior of learning heuristics has previously been studied mostly for the case when all players use the same heuristic. For the case of imitate-the-best, 1 Vega-Redondo =-=[29]-=- showed that in a symmetric Cournot oligopoly with imitators, the long run outcome converges to the competitive output if small mistakes are allowed. This result has been generalized to aggregative qu... |

89 |
Why Imitate, and if so, how? A Boundedly Rational Approach to Multi-arm Bandits
- Schlag
- 1998
(Show Context)
Citation Context ...re contained in our companion paper, Duersch, Oechssler, and Schipper [9]. 13 Apart from close variants of imitate-if-better like rules that imitate only with a certain probability, see e.g. Schlag’s =-=[25]-=- proportional imitation rule. 25Table 1: Summary of results Class Result Reference Examples Symmetric 2x2 games essentially unbeatable Prop. 2 Chicken, Prisoners’ Dilemma, Stag Hunt Additively separa... |

63 |
Cournot Oligopoly and the Theory of Supermodular Games
- Amir
- 1996
(Show Context)
Citation Context ...a finite set, then Proposition 5 implies that imitation is not subject to a money pump. Example 13 Consider a symmetric two-player game with the payoff function given by π(x, y) = x y with x, y ∈ X ⊂ =-=[1, 2]-=- with X being finite. This game’s relative payoff function is quasiconcave. Thus our result implies that imitation is not subject to a money pump. Moreover, the example demonstrates that not every qua... |

56 |
Two Person Cooperative Games." Econometrica 21
- Nash
- 1953
(Show Context)
Citation Context ...ts imply that imitation is not subject to a money pump in bargaining as modeled in the Nash Demand game. Example 11 (Nash Demand Game) Consider the following version of the Nash Demand game (see Nash =-=[17]-=-). Two players simultaneously demand an amount in R+. If the sum is within a feasible set, i.e., x + y ≤ s for s > 0, then player 1 receives the payoff π(x, y) = x. Otherwise π(x, y) = 0 (analogously ... |

51 | Imitation and Belief Learning in an Oligopoly Experiment - Offerman, Potters, et al. - 2002 |

49 |
Evolutionary stable strategies for a finite population and a variable contest size
- Schaffer
- 1988
(Show Context)
Citation Context ... imitation cycle.□ As in previous studies of imitation (see e.g. Alós-Ferrer and Ania [1]; Schipper [24]; Vega-Redondo [29]), the concept of a finite population evolutionary stable strategy (Schaffer =-=[22]-=- [21]) plays a prominent role in our analysis. Definition 4 (fESS) An action x ∗ ∈ X is a finite population evolutionary stable strategy (fESS) of the game (X, π) if π(x ∗ , x) ≥ π(x, x ∗ ) for all x ... |

38 | Learning in Cournot Oligopoly - An Experiment
- Huck, Normann, et al.
- 1999
(Show Context)
Citation Context ...the competitive output if small mistakes are allowed. This result has been generalized to aggregative quasisubmodular games by Schipper [24] and Alós-Ferrer and Ania [1]. Huck, Normann, and Oechssler =-=[12]-=-, Offerman, Potters, and Sonnemans [18], and Apesteguia et al. [4], [5] provide some experimental evidence in favor of imitative behavior. In contrast to the above cited literature, the current paper ... |

24 |
Are profit maximisers the best survivors
- Schaffer
- 1989
(Show Context)
Citation Context ...ation cycle.□ As in previous studies of imitation (see e.g. Alós-Ferrer and Ania [1]; Schipper [24]; Vega-Redondo [29]), the concept of a finite population evolutionary stable strategy (Schaffer [22] =-=[21]-=-) plays a prominent role in our analysis. Definition 4 (fESS) An action x ∗ ∈ X is a finite population evolutionary stable strategy (fESS) of the game (X, π) if π(x ∗ , x) ≥ π(x, x ∗ ) for all x ∈ X. ... |

22 |
Ania A (2005) The Evolutionary Stability of Perfectly Competitive Behavior. Econ Theor 26
- Alos-Ferrer
(Show Context)
Citation Context ... corollary may be useful for applications. Let X ⊂ Rm be a finite subset of a finite dimensional Euclidean space. A function f : X −→ R is convex (resp. concave) if for any x, x ′ ∈ X and for any λ ∈ =-=[0, 1]-=- such that λx + (1 − λ)x ′ ∈ X, f(λx + (1 − λ)x ′) ≤ (≥)λf(x) + (1 − λ)f(x ′). Corollary 7 Let (Rm, π) be a symmetric two-player game for which π(·, ·) is concave in its first argument and convex in i... |

19 |
Rent dissipation in a limited-access commonpool resource: experimental evidence
- Walker, Gardner, et al.
- 1990
(Show Context)
Citation Context ...ven by π(x, y) = c(e − x) + The return from investment x x+y (a(x + y) − b(x + y)2 ), with a, b > 0. So the x x+y (a(x + y) − b(x + y)2 ) if x, y > 0 and ce otherwise (see Walker, Gardner, and Ostrom =-=[30]-=-). Since ∆(x, y) = (c(e − x) + ax − bx 2 ) − (c(e − y) + ay − by 2 ), Proposition 3 implies that imitation is essentially unbeatable. Example 6 (Minimum Effort Coordination) Consider the class of mini... |

15 |
Evolutionary equilibrium in Tullock contests: spite and overdissipation
- Hehenkamp, Leininger, et al.
- 2004
(Show Context)
Citation Context ... x x+y zero if both players bid zero. The cost of bidding equals the bid. The symmetric payoff function is given by π(x, y) = x v − x (see Tullock [27] and Hehenkamp, Leininger, and x+y Possajennikov =-=[11]-=-. This game is an aggregative quasisubmodular game (see Schipper [24, Example 6] and Alós-Ferrer and Ania [1, Example 2] and π(x, y) is concave in x. Thus Proposition 6 implies that imitation is not s... |

13 | Imitation: theory and experimental evidence - Apesteguia, Steffen, et al. |

9 | Imitators and optimizers in Cournot oligopoly
- Schipper
- 2006
(Show Context)
Citation Context ...rimental evidence in Duersch, Kolb, Oechssler, and Schipper [8] we are not aware of any work that deals with this issue. For a Cournot oligopoly with imitators and myopic best reply players, Schipper =-=[23]-=- showed that the imitators’ long run average payoffs are strictly higher than the best reply players’ average payoffs. The article is organized as follows. In the next section, we present the model an... |

9 | 2003): “Submodularity and the evolution of Walrasian behavior
- Schipper
(Show Context)
Citation Context ...not oligopoly with imitators, the long run outcome converges to the competitive output if small mistakes are allowed. This result has been generalized to aggregative quasisubmodular games by Schipper =-=[24]-=- and Alós-Ferrer and Ania [1]. Huck, Normann, and Oechssler [12], Offerman, Potters, and Sonnemans [18], and Apesteguia et al. [4], [5] provide some experimental evidence in favor of imitative behavio... |

8 | Evolutionary stability and Nash equilibrium in finite populations, with an application to price competition - Ania - 2008 |

7 | Imitation and the Evolution of Walrasian Behavior: Theoretically Fragile but Behaviorally Robust. http://papers. ssrn.com/sol3/papers.cfm?abstract_id=1095135, February 2008. CESifo Working Paper Series No - Apesteguia, Huck, et al. |

6 |
Rage against the machines: How subjects play against learning computers, Economic Theory, forthcoming
- Dürsch, Kolb, et al.
- 2009
(Show Context)
Citation Context ...orward looking maximizer in large classes of economically relevant situations? The idea for this paper emerged from a prior observation in experimental data. In Duersch, Kolb, Oechssler, and Schipper =-=[8]-=-, subjects played against computers that were programmed according to various learning algorithms in a Cournot duopoly. On average, human subjects easily won against all of their computer opponents wi... |

5 |
Öchssler (2007), Imitation - Theory and Experimental Evidence, Journal of Economic Theory, forthcoming
- Apestegúıa, Huck, et al.
(Show Context)
Citation Context ...as been generalized to aggregative quasisubmodular games by Schipper [24] and Alós-Ferrer and Ania [1]. Huck, Normann, and Oechssler [12], Offerman, Potters, and Sonnemans [18], and Apesteguia et al. =-=[4]-=-, [5] provide some experimental evidence in favor of imitative behavior. In contrast to the above cited literature, the current paper deals with the interaction of an imitator and a forward looking, v... |

3 | Supermodular games and potential games - Brânzei, Mallozzi, et al. - 2003 |

3 |
A simple rational expectations Keynes-type coordination model, Quarterly
- Bryant
- 1983
(Show Context)
Citation Context ...nbeatable. Example 6 (Minimum Effort Coordination) Consider the class of minimum effort games given by the symmetric payoff function π(x, y) = min{x, y} − c(x) for some cost function c(·) (see Bryant =-=[7]-=- and Van Huyck, Battalio, and Beil [28]). Corollary 3 implies that imitation is essentially unbeatable. Example 7 (Synergistic Relationship) Consider a synergistic relationship among two individuals. ... |

2 | 2012b). Pure strategy equilibria in symmetric two-player zero-sum games
- Duersch, Oechssler, et al.
- 1999
(Show Context)
Citation Context ...e a finite symmetric game with its relative payoff game (X, ∆). If (X, ∆) has no pure equilibrium, then imitation is subject to a money pump. 8Proof. By Theorem 1 in Duersch, Oechssler, and Schipper =-=[9]-=-, (X, ∆) has no symmetric pure equilibrium if and only if it is a gRPS matrix. Thus, if (X, ∆) has no symmetric pure equilibrium, then it is a gRPS game. Hence, by Theorem 1 imitation is subject to a ... |

1 | Pure saddle points and symmetric relative payoff games, mimeo - Duersch, Oechssler, et al. - 2010 |

1 | Overtaking and almost-sure optimality for infinite horizon Markov decision processes - Leizarowitz - 1996 |

1 | Are profit-maximizers the best survivors?, Journal of Economic Behavior and Organization - Schaffer - 1989 |