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Fatal Attraction: Salience, Naivete, and Sophistication in Experimental Hide-and-Seek Games Vincent P. Crawford and Nagore Iriberri University of California, San Diego September, 2004; revised February, 2006

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Hide and Seek Games Hide-and-Seek games are zero-sum two- person two-outcome games with in which one player wins by matching the other's decision and one wins by mismatching. Hide-and-Seek games cleanly model a strategic problem that is central to many economic, political, and social settings, as well as military and security applications.

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Examples Entry games where entry requires a differentiated product and blocking it requires matching the entrant's design Election campaigns in which a challenger can win only by campaigning in a different area than the incumbent Fashion games in which hoi polloi wish to mimic the elite but the elite prefer to distinguish themselves

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Puzzle Zero-sum two-person games are one of game theory's success stories But equilibrium analysis of Hide-and- Seek games is not very helpful in applications

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Two main reasons Hide-and-Seek games are often played without clear precedents; so equilibrium (initially) depends on strategic thinking, which may not follow equilibrium logic Hide-and-Seek games are usually played on cultural or geographic "landscapes" with non-neutral payoffs and/or framing of locations; equilibrium ignores the landscape except as it affects payoffs, but non-equilibrium thinking may respond to it

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Popular culture as data "Any government wanting to kill an opponent…would not try it at a meeting with government officials." (comment on the poisoning of Ukrainian presidential candidate (now president) Viktor Yushchenko, quoted in Chivers (2004)) "…in Lake Wobegon, the correct answer is usually 'c'." (Garrison Keillor (1997) on multiple- choice tests)

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Experiments Rubinstein & Tversky '93 ("RT") Rubinstein, Tversky, & Heller '96 ("RTH") Rubinstein '98-'99 ("R") (collectively "RTH") A subject usually played only one Hide- and-Seek game, with an anonymous partner Multi-game designs suppressed learning and repeated-game effects to elicit initial responses

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RTH's design Typical Seeker's instructions (Hider's analogous): Your opponent has hidden a prize in one of four boxes arranged in a row. The boxes are marked as shown below: A, B, A, A. Your goal is, of course, to find the prize. His goal is that you will not find it. You are allowed to open only one box. Which box are you going to open? ABAA

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Non-neutral framing ABAA Focally labeled End Locations "Least Salient" Location The "B" location is distinguished by its label The two "end A" locations may be inherently salient This gives the "central A" location its own brand of uniqueness as the "least salient" location

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Re-landscaping game theory RTH's design is important as a tractable abstract model of a non-neutral cultural or geographic landscape Like our popular-culture illustrations, their design focuses on non-neutral framing of locations, keeping payoffs neutral Rosenthal, Shachat, and Walker (2003) and others give complementary analyses of Hide-and-Seek with non-neutral payoffs

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Equilibrium RTH's game has a unique equilibrium, in which both players randomize uniformly Expected payoffs: Hider 3/4, Seeker 1/4 Hider/Seeker ABAA A0,11,0 B 0,11,0 A 0,11,0 A 0,1

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Framing effects Equilibrium leaves no room for the labeling or order of locations to influence behavior Yet RTH's subjects' responses deviated systematically from equilibrium in ways that were highly sensitive to framing Many examples, like the Yushchenko and Wobegon quotations, suggest that such deviations are not confined to the lab

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Aggregate choice frequencies in six closely analogous RTH treatments (Table I) 2 analogous to B Different locations for B Player roles reversed

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Analogies between treatments RTH took 2 as analogous to B and 3 to central A Mine treatments have same normal form (payoff matrix) as Treasure treatments with reversed player roles, different extensive form (game tree) –Mine treatments test whether difference in game tree explains why Seekers do better than Hiders –But RTH's results were the same with reversed player roles, suggesting a normal-form explanation (like all of those considered here) We treat Mine treatments as Treasure treatments with reversed roles, and identify 3 with central A

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"Stylized facts" In all six treatments, given the analogies –Central A (or 3) is most prevalent for both Hiders and Seekers –Central A is even more prevalent for Seekers (or Hiders in Mine treatments) As a result, Seekers (or Hiders in Mine treatments) do systematically better than in equilibrium Puzzling because Seekers are surely as smart as Hiders, on average, and Hiders tempted to hide in central A should realize that Seekers will be just as tempted to look there (Poe's The Purloined Letter); the role asymmetry in (2) is even more puzzling

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Aggregate Choice Frequencies (Table 1) "Stylized facts"

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Pooled Aggregate Choices ABAA Hiders (624) Seekers (560) Chi-square tests for aggregate differences in choice frequencies across the six treatments in Table I reveal no significant differences for Seekers (p-value 0.48) or Hiders (p-value 0.16)

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RTH took these patterns as evidence that their subjects did not think strategically "The finding that both choosers and guessers selected the least salient alternative suggests little or no strategic thinking.“ "In the competitive games, however, the players employed a naïve strategy (avoiding the endpoints), that is not guided by valid strategic reasoning. In particular, the hiders in this experiment either did not expect that the seekers too, will tend to avoid the endpoints, or else did not appreciate the strategic consequences of this expectation."

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But subjects may be smarter than they seem... Such robust patterns are unlikely to lack a coherent explanation Given the simplicity of the strategic problem in Hide-and-Seek, the explanation is unlikely to be nonstrategic Zero-sum games, where the rationale for equilibrium is especially strong, may be an especially good place to compare alternative, strategic but non-equilibrium, theories of behavior

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Outline 1. Comparison of two alternative explanations: (A) Equilibrium (and Quantal Response Equilibrium) with payoff perturbations (B) Level-k thinking, a structural non-equilibrium model of initial responses 2. Econometric analysis. 3. Model Evaluation 1: Overfitting test. 4. Model Evaluation 2: Portability to similar games. 5. Conclusions

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(A)Equilibrium with Payoff Perturbations e and f reflect "hard-wired preferences": players get some payoff directly from location chosen, regardless of opponent’s choice Assume the payoff perturbations e > 0 (end locations) and f > 0 (focally labeled), with magnitudes the same for Hiders and Seekers; relaxed below Signs motivated by strategic intuitions about games like Hide-and-Seek –Hiders fear salience so both perturbations enter negatively for them –Seekers are attracted to salience, so both perturbations enter positively for them Hider/Seeker ABAA A0-e, 1+e1-e, 0+f1-e, 01-e, 0+e B1-f, 0+e0-f, 1+f1-f, 01-f, 0+e A1, 0+e1, 0+f0,11, 0+e A1-e, 0+e1-e, 0+f1-e, 00-e, 1+e

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Equilibrium with perturbations of equal magnitudes Continue to assume that the magnitudes of e and f are the same for Hiders and Seekers (econometric estimates e = 0.22, f = 0.20) New mixed equilibrium, still unique and symmetric: A: 1/4 – e/2+ f/4 B: 1/4+ e/2 – 3f/4 A: 1/4+ e/2+ f/4 >1/4 A: 1/4 – e/2+ f/4 Hiders and Seekers both play central A with probability 1/4+ e/2+ f/4 > 1/4 whenever 2e+ f > 0, so the model can explain the prevalence of central A in both roles But the model cannot explain the greater prevalence of central A for Seekers than Hiders without an unexplained difference in the magnitudes of e and f across roles

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Equilibrium with perturbations of different magnitudes Now assume that the magnitudes of e and f are different for Hiders and Seekers (econometric estimates: e H = 0.29, e S = 0.15; f H = 0.25, f S = 0.15) New mixed equilibrium, still unique but now asymmetric A Seeker still finds the treasure with probability > 1/4 Hiders and Seekers still play central A with probability > 1/4 The model can now "explain" the greater prevalence of central A for Seekers, but only by making 2e + f > 0 almost twice as large for Hiders

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QRE with Payoff Perturbations QRE is a generalization of equilibrium in which players' choices are noisy, with choice probabilities increasing with expected payoff, given the distribution of others' choices (a fixed point in choice-distribution space) In applications it is often assumed that choices follow a logistic distribution ("logit QRE"), with dispersion tuned by a precision parameter λ: noise=1/λ Without perturbations, Hide-and-Seek makes QRE choice probabilities coincide with equilibrium for any λ But with perturbations with signs asymmetric across roles as assumed for equilibrium –Equal magnitudes: QRE predicts the opposite asymmetry (hiders choosing central A with more probability) –Different magnitudes: QRE reduces to equilibrium

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Critique Assuming role-asymmetric payoffs, however intuitive in this case, begs the question of why Hiders' and Seekers' responses differ Unrestricted payoff perturbations give the model enough flexibility to explain virtually any pattern of choices, raising concerns about overfitting Tailoring behavioral assumptions so closely to the strategic structure of Hide-and-Seek may reduce the model's portability, the extent to which estimating its parameters in one setting is useful in predicting behavior in other settings

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(B)The Level-k Model We now consider a model with boundedly rational level-k ("Lk") decision rules or "types" The model describes behavior in other settings involving initial responses to games, which should allay the concern that without equilibrium, "anything is possible" –Stahl and Wilson (1994, 1995) –Nagel (1995) –Ho, Camerer, and Weigelt (1998) –Costa-Gomes, Crawford, and Broseta (2001) –Crawford (2003) –Camerer, Ho, and Chong (2004) –Costa-Gomes and Crawford (2004) –Crawford and Iriberri (2005)

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With given probabilities (to be estimated), the same for each player role, each role is filled by one of five Lk types: L0, L1, L2, L3, or L4 (in Hide and Seek the types cycle after L4) Type Lk for k > 0 anchors its beliefs in a naïve L0 type and adjusts them via thought- experiments with iterated best responses –L1 best responds to L0 (with uniform errors) –L2 best responds to L1 (with uniform errors) –Lk best responds to Lk-1 (with uniform errors) The Level-k Model

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Level-k thinking Lk types for k > 0 have accurate models of the game and are rational (though noisy); they depart from equilibrium only in basing their beliefs on a simplified model of other players Level-k thinking yields a workable model of others’ choices while avoiding much of the cognitive complexity of equilibrium analysis: "Basic concepts in game theory are often circular in the sense that they are based on definitions by implicit properties… Boundedly… rational strategic reasoning seems to avoid circular concepts. It directly results in a procedure by which a problem solution is found. Each step of the procedure is simple, even if many case distinctions by simple criteria may have to be made" (Selten (1998))

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The anchoring type Level-0 The key to the models' explanatory power is the specification of the anchoring type L0 We take L0 to be non-strategic as usual, payoff- insensitive and so role-independent in RTH's games, and represent it directly by its choice probabilities on A, B, A, A : (p/2, q, 1-p-q, p/2) A uniform L0 (as in most applications) would make Lk coincide with equilibrium here Proposed model: we allow L0 Hiders and Seekers both to favor the salient locations, to an equal extent: p > 1/2, q > 1/4

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Explaining the stylized facts when p>1/2, q>1/4 (Table 2) Given L0's attraction to salient locations, L1 Hiders choose central A to avoid L0 Seekers and L1 Seekers avoid central A in searching for L0 Hiders For similar reasons, L2 Hiders choose central A with probability between zero and one and L2 Seekers choose it with probability one L3 Hiders avoid central A and L3 Seekers choose it with probability between zero and one L4 Hiders and Seekers both avoid central A For behaviorally plausible type distributions, the model explains the prevalence of central A for Hiders and Seekers and its greater prevalence for Seekers The role asymmetry in behavior follows from Hiders' and Seekers' asymmetric responses to L0's role-symmetric choices, with no unexplained role differences in behavior

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Other Level-0s? Since the Level-0 is the key in the specification of any level-k we also consider alternative specifications, relaxing: –Reaction to salience: level-k where both the Hider and the Seeker avoid salience (p<1/2 and q<1/4) –Role asymmetry (Bacharach & Stahl): Hiders avoiding salience (pH 1/2 and qS>1/4)

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p q 1 Region 4 L1 H: B L1 S: central A L2 H: B & end As L2 S: B L3 H: central & end As L3 S: B & end As L4 H: central A L4 S: central & end As Region 1 L1 H: central A L1 S: B L2 H: central & end As L2 S: central A L3 H: B & end As L3 S: central & end As L4 H: B L4 S: B & end As 3p +2q = 2 p + 2q = 1 Region 6 L1 H: end As L1 S: B L2 H: central & end As L2 S: end As L3 H: B & central A L3 S: central & end As L4 H: B L4 S: B & central A Region 3 L1 H: B L1 S: end As L2 H: B & central A L2 S: B L3 H: B & central A L3 S: B & central A L4 H: end As L4 S: B & central A Region 2 L1 H: central A L1 S: end As L2 H: B & central A L2 S: central A L3 H: B & end As L3 S: B & central A L4 H: end As L4 S: B & end As Region 5 L1 H: end As L1 S: central A L2 H: B & end As L2 S: end As L3 H: B & central A L3 S: B & end As L4 H: central A L4 S: B & central A p = 2q Figure 5: L1's Through L4's Choices as Functions of L0's Choice Probabilities (estimates, p>1/2, q>1/4, p>2q, in region 2)

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A simpler alternative "Hiders feel safer avoiding salient locations, so they are most likely to choose central A; and Seekers know this, so they are also most likely to choose central A." This has two weaknesses, both remedied by our model –It assumes Hiders are systematically more sophisticated than Seekers –It does not explain why Seekers choose central A even more often than Hiders

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2. Econometric analysis Econometrics are just a way to calibrate the models to fit the observed choice frequencies, constraining our discretion and yielding likelihoods that can be used to assess goodness of fit and the costs of parameter restrictions We pool the data from all treatments, obtaining a sample of 624 Hiders and 560 Seekers We use a mixture-of-types model as in Costa- Gomes, Crawford, and Broseta (2001) We do not seek to take a definitive position on the behavioral parameters, which would require much more comprehensive experiments

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Comparison: Equilibrium with perturbations (eH,fH,eS,fS) Equal magnitudes Different magnitudes Level-k: (Impose r=0 and estimate s,t,u,v, ε) Role-symmetric Level-0 that favors salience Role-symmetric Level-0 that avoids salience Role-asymmetric Level-0 that avoids salience for Hiders and favors salience for Seekers

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Mixture of types model Likelihood function: (,, … ) ( for equilibrium ) level-k type frequency probability for player role i of choosing location j given subject i belongs to level-k type number of subjects in role i who chose action j

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Table 3. Parameter Estimates and Likelihoods for the Leading Models in RTH's Games ModelLn LParameter estimates Observed or predicted choice frequencies MSE Pl.ABAA Observed frequenciesH (624 hiders, 560 seekers)S Equilibrium without H Perturbations S Equilibrium with e H ≡ e S = f H ≡ f S = H restricted perturbations S Equilibrium with e H = , f H = e S = , f S = H unrestricted perturbations S Level-k with a role-symmetric r = 0, s = , t = , u = , v = , ε = 0 H L0 that favors salience S Level-k with a role-asymmetric L0 that r = 0, s = 0.66, t = 0.34, ε=0.72; u ≡ v ≡ 0 imposed H favors salience for seekers and avoids it for hidersS Level-k with a role-symmetric r = 0, s = , t = , u = , v = , ε = 0 H L0 that avoids salience S

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Comments on Table 3 In the equilibrium model, the restrictions that payoff perturbations are equal in magnitude for Hiders and Seekers are strongly rejected. Equilibrium with unrestricted perturbations shows the highest fit. Among the level-k models, the one with L0 favoring salience has slightly worse fit but most sensible (hump-shaped) type frequency.

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3. Model Evaluation 1: Overfitting Given these models' flexibility, overfitting is a concern We test for overfitting by re-estimating each model separately for each of the six treatments and using the re-estimated models to "predict" the choice frequencies of the other five treatments We evaluate goodness of fit by mean squared errors (MSE) between predicted and observed choice frequencies

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Table 4. Overall MSEs in RTH's Games ModelsOverall MSE Level-k with symmetric L0 that favors salience Equilibrium with unrestricted perturbations Level-k with symmetric L0 that avoids salience Level-k with asymmetric L0 that avoids salience for Hiders and favors salience for Seekers Even though our proposed level-k model fits slightly worse than each alternative, it has a lower MSE than each alternative but the level-k model with a role- asymmetric L0 with seekers favoring salience and hiders avoiding it, whose error is 10% lower.

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4. Model evaluation 2: Portability Portability is also a concern Test by using the equilibrium with perturbations and level-k models, estimated from RTH's data, to "predict" subjects' initial responses to close relatives of RTH's Hide-and-Seek game –O'Neill's (1987) card-matching game –Rapoport and Boebel's (1992) closely related game Both games raise the same strategic issues as RTH's Hide and Seek game, but with more complex patterns of wins and losses, different framing, and in the latter case five locations

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O'Neill's Card-Matching Game Player1/ Player2 A (Seeker) 2 (Seeker) 3 (Seeker) J (Hider) A (Hider) 0, 11, 0 0, 1 2 (Hider) 1, 00, 11, 00, 1 3 (Hider) 1, 0 0,1 J (Seeker) 0, 1 1, 0

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O'Neill's Card-Matching Game A, 2, and 3 are strategically symmetric, but we keep them separate because equilibrium with perturbations and level-k break the symmetries Equilibrium has Pr{A} = Pr{2} = Pr{3} = 0.2,Pr{J} = 0.4 Initial choice frequencies are –8% A, 24% 2, 12% 3, 56% J for Player 1 –16% A, 12% 2, 8% 3, 64% J for Player 2 So there is no "Ace effect" initially; that must have been a product of learning But there is a "Joker effect" an order of magnitude larger

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Adapting models to O'Neill's Game O'Neill's game is not a Hide-and-Seek game, but player 1 can be viewed as a hider for A, 2, 3 and a seeker for J; and player 2 as reversing these roles The equilibrium model's payoff perturbations are readily adapted to O'Neill's game (a player choosing a salient card for which he is a seeker (hider) receives (loses) an additional payoff α > 0 for A and ι > 0 for J) Level-k models are easily adapted: (a (1-a-j)/2 (1-a-j)/2 j) –Level-k with symmetric L0 that favors salience: a and j>1/4 –Level-k with symmetric L0 that avoids salience: a and j<1/4 –Level-k with asymmetric L0: a1 1/4; a2 > 1/4, j2 < ¼ Level-k with symmetric L0 that favors salience shows the highest fit among all models.

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Table 5. Comparison of the Leading Models in O'Neill's Game ModelParameter estimatesObserved or predicted choice frequenciesMSE PlayerA23J Observed frequencies (25 Player 1s, 25 Player 2s) Equilibrium without perturbations Level-k with a role-symmetric a > 1/4 and j > 1/ L0 that favors salience 3j – a < 1, a + 2j < Level-k with a role-symmetric a > 1/4 and j > 1/ L0 that favors salience 3j – a Level-k with a role-symmetric a < 1/4 and j < 1/ L0 that avoids salience Level-k with a role-asymmetric L0 that favors salience for locations for which a 1 1/4; a 2 > 1/4, j 2 < 1/ player is a seeker and avoids it for locations for which player is a hider 3j 1 - a 1 < 1, a 1 + 2j 1 < 1, 3a 2 + j 2 >

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Rapoport and Boebel's Game Player1/ Player2 C (Hider) LFIO C (Seeker) 1, 00, 1 L 1, 0 F0, 11, 00, 1 1, 0 I0, 11, 00, 11, 00, 1 O 1, 0 0, 1

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Rapoport and Boebel's Game F, I, and O are strategically symmetric, but we keep them separate Equilibrium has Pr{C} = 0.375, Pr{L} = 0.25, Pr{F} = Pr{I} = Pr{O} = for Players 1 and 2 Initial choice frequencies are –10% C, 5% L, 15% F, 60% I, 10% O for Player 1 –50% C, 30% L, 10% F, 5% I, 5% O for Player 2

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Adapting models to Rapoport and Boebel's game It does not make a player unambiguously a hider or seeker depending on which location he chooses (Player 1 (2) is a seeker (hider) for location C; When C is eliminated, player 1 (2) is a hider (seeker) for location L; but even when L and C are eliminated, the player roles for location F cannot be classified this way) –There is no plausible, parsimonious way to adapt the payoff perturbations model to this game –There is no plausible way to adapt the level-k model with asymmetric L0 to this game The level-k with symmetric L0 models adapt easily and the level-k with symmetric L0 that favors salience fits their data best overall but only slightly better than equilibrium (equilibrium is best for player 2s in treatment 2)

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Table 6. Comparison of the Leading Models in Rapoport and Boebel's Game Model Parameter Estimates Observed or predicted choice frequencies MSE Tr 1 MSE Tr 2 Pl.CLFIO Observed frequencies, Tr (10 Player 1s, 10 Player 2s) Observed frequencies, Tr (10 Player 1s, 10 Player 2s) Equilibrium without perturbations Level-k with a role-symmetric m>2/5, n>1/ L0 that favors salience 3m/2 + n > Level-k with a role-symmetric m>2/5, n>1/ L0 that favors salience 3m/2 + n < Level-k with a role-symmetric m<2/5, n<1/ L0 that avoids salience 2m + 3n < Level-k with a role-symmetric m<2/5, n<1/ L0 that avoids salience 2m + 3n >

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5. Conclusions Both level-k and equilibrium with perturbations models are flexible enough to fit RTH's data very well Level-k with symmetric L0 that favors salience fits slightly worse than alternative models but has comparative advantages on: –More sensible (hump-shaped) type estimates –Overfitting: does better in within–sample "predictions" in RTH's games –Portability to similar games (O’Neill and Rapoport and Boebel): more adaptable and more accurate beyond- sample in O'Neill's and Rapoport-Boebel's games

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Conclusions (continued) The level-k model with L0 that favors salience has several further advantages –Its assumptions seem behaviorally more plausible and it (alone) does not rely on unexplained role differences in behavior or payoffs. –It is based on general decision rules or "types" that apply to any game –Its L0 is based on simple principles—how salience is determined by the set of decisions and their framing, and how people respond to it—for which there is strong support, whose simplicity facilitates transfer to new games, just as the sensitivity to the details of the structure of alternative specifications of L0 or the payoff perturbations in an equilibrium model inhibit transfer.

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ADDITIONAL TABLES AND GRAPHS: –QRE (with equal magnitude and different magnitude perturbations). –Overfitting Appendix: treatment by treatment estimation.

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QRE with perturbations of equal magnitudes (Figure 3) With perturbations of equal magnitudes (but opposite signs) for Hiders and Seekers, logit QRE for reasonable λ is consistent with the prevalence of central A for both Hiders and Seekers But it robustly predicts that central A is more prevalent for Hiders than Seekers: exactly reversing the pattern in RTH's data

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QRE with perturbations of different magnitudes (Figure 4) QRE can only explain the greater prevalence of central A for Seekers via a large, unexplained difference in the magnitudes of the payoff perturbations across Hiders and Seekers For sufficiently high λ the parameters are not identified econometrically, but maximum likelihood estimates all yield the same predictions as equilibrium with perturbations

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Table A2. Treatment by Treatment Parameter Estimates in RTH's Games TreatmentLevel-k with symmetric L0 favoring salience Equilibrium with unrestricted perturbations rstuveeHeH fHfH eSeS fSfS RTH RT-AABA-Treasure RT-AABA-Mine RT-1234-Treasure RT-1234-Mine R-ABAA Treatment Level-k with symmetric L0 avoiding salienceLevel-k with asymmetric L0 rstuverstε RTH RT-AABA-Treasure RT-AABA-Mine RT-1234-Treasure RT-1234-Mine R-ABAA

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