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A quantal response equilibrium model of order-statistic games
Journal of Economic Behavior & Organization Vol. 51 (2003) 413–425
A quantal response equilibrium model of order-statistic games Kang-Oh Yi* Department of Economics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Received 24 August 2001; accepted 15 November 2001
Abstract This paper applies McKelvey and Palfrey’s [Games Econ. Behav. 10 (1995) 6] notion of “quantal response equilibrium” to a class of “order-statistic” coordination games closely related to those studied by Van Huyck et al. [Am. Econ. Rev. 80 (1990) 234; Quart. J. Econ. 106 (1991) 885] and Van Huyck et al. [Evidence on Learning in Coordination Games. Manuscript, Texas A&M University, 1996] and those recently analyzed by Anderson et al. [Games Econ. Behav. 34 (2001) 177]. With quadratic effort costs, the limiting QRE as the noise goes to zero in their games is the most efficient equilibrium. This result contrasts with Anderson et al.’s conclusion for order-statistic games with linear effort costs, and allows a fuller assessment of the QRE’s success in describing the limiting outcomes in Van Huyck et al.’s experiments. © 2002 Elsevier Science B.V. All rights reserved. JEL classification: C79; C92 Keywords: Coordination game; Equilibrium selection; Quantal response equilibrium
1. Introduction McKelvey and Palfrey’s (1995) notion of “quantal response equilibrium” (QRE) was proposed as a way to allow realistic levels of uncertainty about players’ strategies in noncooperative game-theoretic analyses, while retaining most of the parsimony of standard equilibrium analyses. In a QRE, players do not always choose the strategy with the highest expected payoff, as in standard analyses. Instead their strategy choices are noisy, and strategies with higher payoffs are chosen with higher probabilities. As in standard equilibrium analyses, players take the noise in each other’s strategies rationally into account, correctly predicting the distributions of others’ strategies in evaluating the expected payoffs of their own strategies. ∗ Tel.: +852-2358-7619; fax: +852-2358-2084. E-mail address: [email protected] (K.-O. Yi).
0167-2681/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 6 8 1 ( 0 2 ) 0 0 0 9 5 - 1
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In applications of QRE, the noise in players’ strategy choices follows a specific distribution—often the logit distribution familiar from analyses of individual decisions—which allows the degree of noisiness to be represented by as few as one parameter. A QRE with a logit response function is called a logit equilibrium. Logit equilibrium gives a parsimonious description of imperfectly optimizing strategic behavior, which is well suited to the econometric analysis of experimental data. In some applications the noise parameter is estimated and the resulting logit equilibrium is compared with subjects’ observed choices period by period. In others, a limiting logit equilibrium, the limit of logit equilibrium as the noise approaches zero, which is usually an equilibrium in the game without noise, is compared with limiting behavior in the experiment. McKelvey and Palfrey (1995, 1998) and others have shown that the logit equilibrium is surprisingly successful in describing the quantitative as well as qualitative patterns of deviation from equilibrium observed in a variety of game experiments. Logit equilibrium also frequently implies particular patterns of selection among multiple equilibria, suggesting that it may be useful in describing the outcomes of coordination experiments. These studies have established the logit equilibrium as a standard tool in applied game theory and reserved a place for it on the list of potentially useful models of equilibrium selection. Although many qualitative properties of QRE have been established by general theoretical arguments (see for example Chen et al. (1997) and Anderson et al. (1998)), it is difficult to derive its quantitative implications analytically because it is defined as a fixed point in the space of distributions of players’ strategies. As a result, most quantitative applications of QRE have relied on numerical analyses. For similar reasons, little is known about the implications of QRE in coordination games. A notable exception on both counts is Anderson et al. (2001) (AGH), who have recently given a quantitative analytical characterization of QRE for an important class of “order-statistic” coordination games with Pareto-ranked equilibria. The games AGH studied and the games studied in this paper are closely related to those used in the classic experiments of Van Huyck et al. (1990, 1991) (VHBB) and Van Huyck et al. (1996) (VHBB). To understand AGH’s results and the results of the present paper, it is necessary to describe VHBB’s and VHBR’s games and results in more detail.1 In VHBB’s and VHBR’s games, each of n players chooses simultaneously among pure strategies called “efforts,” and players’ payoffs and best responses are determined by their own efforts and an order-statistic of their own and other players’ efforts. In VHBB (1990) the order-statistic was either the minimum of the efforts of the whole group of 14–16 (treatment A) or of pairs randomly selected from the whole group (treatment Cd ), and there was a linear payoff penalty for efforts above the minimum. In a variant of the “large-group” treatment A, the cost of effort was lowered to 0, making playing for the efficient equilibrium a weakly dominant strategy (treatment B). In VHBB (1991) the order-statistic was the median of the whole group of nine, and in the leading such treatment (Γ ) there was a quadratic penalty for efforts different from the median, which I will refer to as “quadratic effort costs” (abusing terminology somewhat because the penalty is symmetric for positive and negative deviations from the ideal effort). In VHBR’s experiments, either five or seven subjects chose among 101 efforts, the order-statistic being either the second or the fourth from the lowest effort among 1
My description follows Crawford (1991, 1995), which provides more detail on VHBB’s design and results.
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five or seven effort choices, and there was again a quadratic penalty for efforts different from the order-statistic. In each case, any symmetric effort combination is an equilibrium, and these equilibria are Pareto-ranked; but there is a tension between the higher payoffs of the Pareto-efficient equilibrium and its greater fragility, which makes it riskier to play for the efficient equilibrium when others’ responses are not perfectly predictable. These games capture important aspects of coordination problems in economic environments and resemble a number of economic models, including the models of Keynesian coordination failure in Bryant (1983) and Cooper and John (1988). Although VHBB’s and VHBR’s games all have similar structures from a game-theoretic point of view, the dynamics and limiting outcomes varied across treatments in ways that shed light on how the strategic setting affects the efficiency of coordination and that discriminate sharply among alternative models of strategic behavior. In VHBB’s (1990) treatment A, subjects converged rapidly to the least efficient equilibrium; while in treatment B, subjects converged rapidly to the most efficient equilibrium; and in treatment Cd , the outcomes were moderately inefficient but converged very slowly. In VHBB’s (1991) large-group median treatments, by contrast, subjects always locked in on the equilibrium determined by the initial median of their efforts, even though it varied across runs and was usually inefficient. In VHBR’s treatments, the results were highly heterogeneous across runs even within a treatment, sometimes converging to the efficient equilibrium and sometimes not. Thus, the limiting outcomes in VHBB’s experiments varied systematically with the number of subjects, their interaction pattern, the order-statistic, the penalty function, and the coarseness of the strategy spaces in ways that provide an interesting question for analysis. AGH’s analysis, like the present paper’s, restricts attention to games with a bounded continuum of efforts, approximating VHBR’s 101-effort games but with strategy spaces significantly less coarse than in VHBB’s original seven-effort games. AGH showed, for minimum games with linear effort costs and payoff parameter values like VHBB’s, that the limiting logit equilibrium always selects a unique equilibrium in the game without noise. This equilibrium is the least efficient equilibrium in VHBB’s treatment A, the most efficient equilibrium in treatment B, and the equilibrium with one-half of the maximum possible effort in treatment Cd . The limiting logit equilibrium therefore varies across VHBB’s (1990) minimum treatments in ways that allow it to describe the limiting experimental outcomes with surprising accuracy. AGH’s results suggest that logit equilibrium model might also be useful in describing the limiting outcomes of VHBB’s (1991) and VHBR’s experiments. VHBB’s (1991) games differ from those studied by AGH in having coarser strategy spaces (with seven efforts as in VHBB (1990)), order-statistics different than the minimum, and quadratic rather than linear effort costs. VHBR’s games have strategy spaces that approximate the bounded continuum of AGH’s model, but differ from AGH’s games in the last two ways. AGH’s analysis leaves open the question of whether the limiting logit equilibrium can also describe the very different limiting outcomes in these experiments, and if so, which of the differences just mentioned accounts for the different limiting outcomes. This paper supplements AGH’s results by providing a general, quantitative characterization of the limiting logit equilibrium in another important subclass of order-statistic games with a bounded continuum of efforts, those with quadratic effort costs, as in VHBB’s (1991)
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and VHBB’s (1996) games.2 The main result is that, for all such games, no matter what the order-statistic is, the limiting logit equilibrium is the Pareto-efficient equilibrium. Thus, AGH’s result for large-group minimum games with linear effort costs is reversed for quadratic effort costs. The analysis shows that this is a natural consequence of the near-neutrality of the logit equilibrium in order-statistic games with a continuum of efforts and the “small” (locally negligible) cost of deviating from an equilibrium with quadratic effort costs. This characterization makes it possible to compare the limiting logit equilibrium with a much more varied set of experimental outcomes. Overall, my results suggest that the limiting logit equilibrium is significantly less successful in describing the limiting outcomes of VHBB’s (1991) and VHBB’s (1996) experiments with median and other order-statistic games with quadratic effort costs than in VHBB’s (1990) minimum games with linear effort costs.3 My results also illustrate the delicacy of logit equilibrium analysis in coordination games with fine strategy spaces, highlighting the importance of seemingly minor details of the cost function. I close by briefly discussing subjects’ incentives, in games with fine strategy spaces and quadratic effort costs, to use small variations in an effort to signal a willingness to move to a more efficient equilibrium. This suggests a partial explanation of the otherwise puzzling outcomes of VHBR’s experiment, in the spirit of Camerer et al. (2002) analysis of “strategic teaching.” The rest of the paper is organized as follows. Section 2 introduces a class of order-statistic games with bounded, continuous strategy spaces and characterizes their logit and limiting logit equilibria. Section 3 studies subjects’ incentives to signal in games with fine strategy spaces and quadratic effort costs, suggesting a partial explanation of VHBR’s results. Section 4 is the conclusion. Proofs omitted from the text are in Appendix A.
2. Order-statistic games and logit equilibrium In an n-person jth order-statistic game, each player chooses an effort level xi ∈ [0, x¯ ], i = 1, . . . , n, where x¯ is a finite maximum effort level. Each player is assumed to be risk-neutral. Let ui (xi , mj:n ) be the player i’s payoff when he plays xi and the resulting order-statistic is mj:n . In this paper, I use a specific functional form which has been used in Van Huyck et al. (1991, 1996). ui (xi , mj:n ) = amj:n − b(mj:n − xi )2 + c,
a, b, c ≥ 0,
(1)
where mj:n is the jth inclusive order-statistic which is defined by m1:n ≤ m2:n ≤ · · · ≤ mn:n , where the mj:n is the jth element of choice combinations {x1 , . . . , xn } arranged in increasing order. In QRE, the probability density of player i’s choosing xi is a function of the expected payoff πie (xi ) and the density of each choice is an increasing function of the expected payoff 2 My research for the first version of this paper (Yi, 1999) was independent of AGH’s work, although the current version has benefited significantly from their analysis. 3 Crawford (1995) proposes an alternative explanation of Van Huyck et al.’s (1990, 1991) results, which also misses some important features of VHBR’s results.
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for that choice. In this paper, I focus on a specialized version of QRE where the choice probabilities are an analogue of the standard multinomial logit distribution as in AGH. fi (xi ) = x¯ 0
exp(λπie (xi ))
exp(λπie (y)) dy
,
(2)
where 0 ≤ λ < ∞ measures the amount of noise, or equivalently, the degree of rationality. This functional form is called a logit function where the odds are determined by the exponential transformation of the utility times a given non-negative constant λ.4 As λ → ∞, the probability of the choice having the highest expected payoff becomes 1, if it is unique, so that the choice behavior becomes best response; when λ = 0 all choices have equal probability. Logit equilibrium for a given λ is defined by a fixed point in these probability distributions. In other words, in equilibrium players’ logit response functions are mutually “noisy best responses,” given λ. To apply the probabilistic choice rule, Eq. (2), to order-statistic games, we need to compute expected payoff, πie (xi ). Let Fi (x) denote the cumulative distribution function associated with fi (x). Let Gij:n−1 (x) be the cumulative distribution function of jth order-statistic regarding {x1 , . . . , xi−1 , xi+1 , . . . , xn } where the xs are drawn from distributions, F−i = {F1 , . . . , Fi−1 , Fi+1 , . . . , Fn }, respectively. That is, Gij:n−1 (x) =
n−1 k k=j Sk l=1
Fil (x)
n−1
(1 − Fil (x)),
l=k+1
where the summation Sk extends over all permutations i1 , . . . , in−1 of 1, . . . , n − 1 for which i1 < · · · < ik and ik+1 < · · · <in−1 . When j = 1, this can be simplified as Gi1:n−1 (x) = 1 − i=k (1 − Fi (x)) where i=k Fi (x) = F1 (x) · · · Fk−1 (x) · · · Fk+1 (x) · · · i (x) be the associated probability density function. Since the logit effort Fn (x). Let gj:n−1 density is invariant to the changes in the origin, the expected payoff can be normalized by excluding those terms that are independent of xi . The normalized expected payoffs in order-statistic games are given in Lemma 1. Lemma 1. In an n-person minimum game, the part of expected payoff that is relevant to logit equilibrium (the part that depends on xi ) is xi xi i i πi (xi ) = a (1 − G1:n−1 (y)) dy + 2b (y − xi )G1:n−1 (y) dy . (3) 0
0
In an n-person order-statistic game with 2 ≤ j ≤ n − 1, the part of expected payoff that is relevant to logit equilibrium is πi (xi ) = 2bEi (mj−1:n−1 )xi − bx2i xi + (a − 2by + 2bxi )(Gij−1:n−1 (y) − Gij:n−1 (y)) dy.
(4)
0
4 In principle, logit equilibrium permits players to have different λs, but the common knowledge assumption is indispensible for the present analysis.
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Substituting πi (xi ) into Eq. (2) completes the determination of the effort densities in order-statistic games. For finite normal-form games, McKelvey and Palfrey (1995) proved the existence and the convergence of logit equilibrium to a Nash equilibrium as λ goes to infinity. In order-statistic games, the same properties hold with continuous strategy spaces. Proposition 1. In any order-statistic game with quadratic effort costs, there exists a logit equilibrium for every λ ≥ 0. The proof of Proposition 1 is a simple modification of the existence proof in Anderson et al. (1998). For a complete proof, see Yi (1999). Given the existence of logit equilibrium, since infinite λ is the “rationality” limit, a limiting logit equilibrium gives a selection from the set of Nash equilibria (McKelvey and Palfrey, 1995). In general, logit equilibrium and limiting logit equilibrium are not unique. In particular, in finite games with multiple strict equilibria, any strict Nash equilibrium can be found as a limit of logit equilibrium.5 But Proposition 2 shows that the limiting logit equilibrium is unique in the order-statistic games studied here. Proposition 2. In any order-statistic game with quadratic effort costs, as λ goes to infinity, the logit equilibrium converges to the most efficient Nash equilibrium, xi = x¯ for all i. Proposition 2 shows that, with quadratic effort costs, as long as a > 0 the limiting logit equilibrium is always efficient regardless of the order-statistic, the number of players, and the values of payoff parameters. The intuition behind this result is that, at xi∗ = Ei (mj:n |xi∗ ), the expected benefit to an increase in one’s own effort is linear but the cost is quadratic, so for a sufficiently small increase in one’s effort the benefit always exceeds the cost. This small “tilt” in favor of higher efforts allows the limiting logit equilibrium with quadratic effort costs to be very different than the limiting logit equilibrium with linear effort costs, even when the order-statistic is the minimum. To see the relation between the benefit and cost of increasing effort more clearly, consider the following payoff function. ui (xi , mj:n ) = amj:n − b(mj:n − xi )2 + c − dxi ,
d ≥ 0.
The associated expected payoff is πie (xi ) = aEi (mj:n |xi ) − bEi [(mj:n |xi ) − Ei (mj:n |xi )]2 + c − dxi = aEi (mj:n |xi ) − bVar i (mj:n |xi ) + c − dxi , where Var(·) denotes the variance. Its first derivative with respect to xi is ∂πie (xi ) ∂Ei (mj:n |xi ) ∂Var i (mj:n |xi ) =a −b − d. ∂xi ∂xi ∂xi
(5)
5 By definition, a strict equilibrium is necessarily a pure-strategy equilibrium. Strict equilibria remain strict when the payoffs are slightly perturbed.
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Given other players’ strategies, the first term on the right-hand side is strictly positive as long as λ is finite and the second term is zero at xi∗ , where the variance attains its minimum. Therefore, the value of a(∂Ei (mj:n |xi∗ )/∂xi ) − d indicates the direction of the best response. When d = 0, the best response effort level is always higher than xi∗ , and there is a force that pushes the effort density towards to the maximum effort level. An interior equilibrium effort level is possible only when the values of a and d preserve a delicate balance between the marginal benefit and cost. For generic values of a and d, a limiting logit equilibrium effort level will be at one or the other extreme. Note that this argument depends only on the quadratic effort costs, not on the details of the noise structure. Thus noise-based notions of equilibrium in order-statistic games with “locally small” effort costs can be expected to tip the balance of the benefit and the cost in favor of a more efficient equilibrium more generally. When a = 0, the expected payoff is πie (xi ) = −bEi (mj:n − xi )2 + c. In this case, the expected payoff is maximized at xi∗ = E(mj:n |xi∗ ). As a result, the limiting logit equilibrium depends on the statistical properties of the order-statistic via j and n. For instance, in a minimum game, since effort level 0 is a unique weakly dominant strategy and there is no mixed-strategy Nash equilibrium, the limiting logit equilibrium should be xi = 0 for all i as λ goes to infinity. The proof for other order-statistic games with a = 0 is quite involved and tedious, and is omitted. See Yi (1999) for the complete proof. Proposition 3. In an order-statistic game with a quadratic payoff function with a = 0, the limiting logit equilibrium for j < (n/2) is xi = 0 for all i; and for j > (n/2) + 1, it is xi = x¯ for all i.
3. Logit equilibrium and experimental results Although the results in AGH and the previous section show that the limiting logit equilibrium gives a unique prediction in order-statistic games, it fails to match some of the experimental results. As AGH showed, VHBB’s (1990) experimental results for minimum games with linear effort costs correspond closely to the limiting logit equilibrium in each of VHBB’s treatments A, B, and Cd . However, VHBB’s (1991) and VHBB’s (1996) experimental results differ from the efficient limiting logit equilibrium identified in Proposition 2. This section explains in more detail how Proposition 2 relates to VHBB’s (1991) and VHBB’s (1996) results and how the coarseness of the strategy space affects players’ incentives. In VHBB’s (1991) experiment, nine subjects played a median game with Xi = {1, 2, . . . , 7} and d = 0. In treatment Γ , a = 0.10 and b = 0.05, and treatment Φ used a = 0 and b = 0.05.6 In both treatments, subjects’ efforts invariably converged by the last period to the initial median, which was effort 4 in some runs and effort 5 in others. 6 There was also a treatment, Ω, where a player’s payoff increases linearly in the median effort, but the payoff is 0 when his effort is different from the order-statistic. Since the payoff structure is very different from one analyzed here, treatment Ω is not considered.
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In VHBR’s experiment, either five or seven subjects played a second- or fourth-orderstatistic game with 101 effort levels, and with the identical payoff parameters used in treatment Γ in VHBB (1991). The strategy space was ei ∈ {0, 1, . . . , 100}, and the map used to determine payoffs was xi = 1 + 0.06ei . Such a transformation brings the environment much closer to the continuum of efforts of AGH’s and my models. It also led to results that differed from VHBB’s (1991) in surprising ways. In VHBR’s experiment, the play converged to the efficient equilibrium in 15 out of 36 runs, with significantly less history dependence than appeared in VHBB (1991). The dynamics suggest that may subjects focused on the current order-statistic and then engaged in local experimentation that are skewed upward, in the direction of efficiency.7 The fact that this experimentation occurred in VHBR’s but not VHBB’s (1991) experiment may be due to the effect of VHBR’s finer strategy space on the cost of small experimentation. Letting mj:n (x1 , . . . , xn ) be an order-statistic and ei = α + βxi with β ≥ 0, we have mj:n (e1 , . . . , en ) = α + βmj:n (x1 , . . . , xn ). From Eq. (5), transforming the strategy space from xi to ei gives %πie (ei ) %Ei (mj:n | ei ) %Var i (mj:n | ei ) =β a − βb −d . (6) %ei %ei %ei When player i raises the effort by one unit, on average, the relative effort cost to the benefit is reduced by 16 times more in VHBB’s (1996) games than in VHBB’s (1991) game. Given this observation, it is natural to ask why AGH’s analysis of limiting logit equilibrium with a continuum of efforts was so successful in predicting the limiting outcomes in VHBB’s (1990) treatments with coarse effort spaces. Recall that VHBB’s (1990) minimum game experiment used essentially three different treatments A, B, and Cd with Xi = {1, 2, . . . , 7}. Treatments A and B are played by 14-16 players, and two players, randomly paired with new partners each period, played each game in treatment Cd .8 In treatments A and Cd , a = 0.20, b = 0, and d = 0.10, while in treatment B, d = 0. This difference may be due to the fact that, since b = 0 for all treatments, AGH’s continuous approximation to the seven-effort games barely affects players’ cost of experimentation in contrast to the strong effect in Eq. (6). In each treatment, the dynamics yield a limiting outcome surprisingly close to the limiting logit equilibrium that is xi = 0 if d > a/n; xi = (¯x/n) if d = (a/n); and xi = x¯ if d < (a/n) for all i (AGH). This difference may explain some, but not all, of the difference between VHBB’s (1991) and VHBB’s (1996) results. It appears that a model in which players focus on stagegame strategies is not enough for a full explanation. Along this line of research, there are some potential explanations connecting history dependence and grid size. VHBR suggested that a fine action grid lowers the opportunity cost of exploring risky but mutually 7 The average change in effort choices is small ranging from −0.54 to 2.62, and in all but one run the value is positive. 8 In the random-pairing treatment C each subject was told only his current pair’s minimum. There was also d a treatment, Cf , in which the minimum game was repeatedly played by fixed pairs, with subjects informed that their pairing were fixed. As Crawford (1995) notes, those subjects were evidently aware of the importance of repeated-game strategies and usually achieved efficient coordination. Because repeated-game strategies raise difficult new issues, that treatment is not discussed here.
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beneficial strategies and thus facilitates coordination on a better outcome. Camerer et al. (2002) proposed a model of “strategic teaching,” in which players choose strategies with poor short-run payoffs expecting better long-run payoffs when there are enough “sophisticated” players, which might provide a way to formalize VHBR’s suggested explanation.
4. Concluding remarks This paper provides a general characterization of logit equilibrium in a class of orderstatistic games with a bounded continuum of efforts and quadratic effort costs, supplementing AGH’s (2001) analysis of similar games with linear effort costs. The results show that the limiting logit equilibrium is significantly less successful in describing VHBB’s (1991) and VHBB’s (1996) results than VHBB’s (1990) results. They also highlight the subtlety of players’ incentive problems in order-statistic games with respect to the coarseness of strategy spaces and the functional form of cost of effort.
Acknowledgements I am deeply grateful to Vincent Crawford and Joel Sobel for their advice and encouragement.
Appendix A Proof of Lemma 1. In a minimum game, xi i πie (xi ) = (ay − b(y − xi )2 )g1:n−1 (y) dy + axi (1 − Gi1:n−1 (xi )) + c 0 xi xi Gi1:n−1 (y) dy + 2b yGi1:n−1 (y) dy = axi Gi1:n−1 (xi ) − a 0 0 xi − 2bxi Gi1:n−1 (y) dy + axi − axi Gi1:n−1 (xi ) + c. 0
Since the logit effort density is invariant to the changing of the origin, c is irrelevant in determining the equilibrium density function. When 2 ≤ j ≤ n − 1, given xi , P(mj:n < y) = P(mj:n−1 < y) = Gij:n−1 (y),
for y < xi
P(mj:n < y) = P(mj−1:n−1 < y) = Gij−1:n−1 (y),
for y > xi
P(mj:n = y) = P(mj−1:n−1 ≤ y ≤ mj:n−1 ) = Gij−1:n−1 (y) − Gij:n−1 (y),
for y = xi .
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Then the expected payoff becomes xi ygij:n−1 (y) dy − b πie (xi ) = a 0
x¯
0
i (y − xi )2 gj:n−1 (y) dy
x¯
x¯
i (y − xi )2 gj−1:n−1 (y) dy xi xi xi i i i +ax(Gj−1:n−1 (xi ) − Gj:n−1 (xi )) + c = −a y(gj−1:n−1 (y) 0 xi i i i (y)) dy + b (y − xi )2 (gj−1:n−1 (y) − gj:n−1 (y)) dy −gj:n−1
+a
+a 0
ygij−1:n−1 (y) dy
xi
−b
0
x¯
ygij−1:n−1 (y) dy − b
0
i (y − xi )2 gj−1:n−1 (y) dy
+ax(Gij−1:n−1 (xi ) − Gij:n−1 (xi )) + c = −axi (Gij−1:n−1 (xi ) xi i −Gj:n−1 (xi )) + a (Gij−1:n−1 (y) − Gij:n−1 (y)) dy + bx2i (Gij−1:n−1 (xi ) 0 xi y(Gij−1:n−1 (y) − Gij:n−1 (y)) dy −Gij:n−1 (xi )) − 2b 0 xi 2 i (Gij−1:n−1 (y) −2bxi (Gj−1:n−1 (xi ) − Gij:n−1 (xi )) + 2bxi
0 i 2 i i −Gj:n−1 (y)) dy + bxi (Gj−1:n−1 (xi ) − Gj:n−1 (xi )) + aEi (mj−1:n−1 ) −bEi (m2j−1:n−1 ) + 2bxi Ei (mj−1:n−1 ) − bx2i + axi (Gij−1:n−1 (xi ) −Gij:n−1 (xi )) + c = aEi (mj−1:n−1 ) − bEi (m2j−1:n−1 ) + 2bEi (mj−1:n−1 )xi xi −bx2i + (a − 2by + 2bxi )(Gij−1:n−1 (y) − Gij:n−1 (y)) dy + c. 0
Since the logit effort density is invariant to the changing of the origin, the first two terms in the last equation are irrelevant in determining the equilibrium density function as well as c. Proof of Proposition 2. Given existence, Theorem 2 and Lemma 2 in McKelvey and Palfrey (1995) show that any limiting logit equilibrium is a Nash equilibrium. Having the convergence result, the proof consists of two parts. After I show that any logit equilibrium is symmetric across players and every symmetric Nash equilibrium is a pure-strategy equilibrium, I show that logit equilibrium selects the Pareto-efficient equilibrium. The first part of the proof concerns only a minimum game. Basically the similar argument applies to the other order-statistic games, and for a complete proof see Yi (1999). To show symmetry by a contradiction, suppose that player 1 and 2have different logit equilibrium densities. Although they have different effort densities, they should hold the beliefs about the other (n − 2) players’ strategies. Let Qj:n−2 (x) be the probability that exactly j of the xi s are less than or equal to x and n − j − 2 are greater than or equal to x in n − 2 random
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samples drawn from {F3 , . . . , Fn }. Then, G1j:n−1 (x) = Qj−1:n−2 (x)F2 (x), and G2j:n−1 (x) = Qj−1:n−2 (x)F1 (x), where Q0:n−2 (x) =
n−2
(1 − Fi (x)).
i=3
Without loss of generality, since f1 (x) and f2 (x) should cross at least once, f1 (x) > f2 (x) and F1 (x) > F2 (x) on (xa , xb ) and f1 (xb ) = f2 (xb ), where xa is the smallest one from which two densities have different values and xa could be 0. Let H(x) ≡ (1/λ)((f1 (x)/f1 (x)) − (f2 (x)/f2 (x))) = Dx π1 (x) − Dx π2 (x).9 Then we have H(xb ) < 0, or xb G21:n−1 (y) − G11:n−1 (y) dy H(xb ) = a(G21:n−1 (xb ) − G11:n−1 (xb )) + 2b 0 xb (F1 (y)−F2 (y))Q0:n−2 (y) dy < 0. = a(F1 (xb ) − F2 (xb ))Q0:n−2 (x) + 2b 0
However, this contradicts F1 (x) > F2 (x) for all x ∈ {xa , xb }. To show that any symmetric equilibrium is pure by a contradiction, suppose that there is a mixed-strategy Nash equilibrium that could have three forms, either the effort levels played with positive probability are isolated or the support involves contiguous intervals or both. Consider the possibility of a contiguous interval first. Let xm and xm be two effort levels in the same interval such that xm < xm . If the support of an equilibrium strategy involves an interval [xm , xm ], then a player’s expected payoff must be constant over the interval and the first derivative of expected payoff exists and must be zero as well as the second derivative. For x ∈ [xm , xm ], however, Dx2 πe (x) = −(a + 2b)g1:n−1 (x) < 0, which is a contradiction. Hence, the equilibrium density can involve only atoms. Now suppose that more than one effort level are played with a positive probability. Let xa denote the minimum effort level and let xb be the second largest effort level being played with a positive probability. Then the expected payoff from xa is πie (xa ) = axa + c, which should be greater than or equal to the expected payoffs for all efforts x∗ ∈ (xa , xb ), πie (x∗ ) = p[axa − b(xa − x∗ )2 ] + (1 − p)ax∗
= axa − bp(xa − x∗ )2 + a(1 − p)(x∗ − xa ),
9 Logit equilibrium effort density f (x ) is jointly continuous in x and λ, and differentiable with respect to x . i i i i It is because πi (xi ) is jointly continuous in xi and λ, and differentiable with respect to xi , and fi (xi ) is the ratio of continuous transformation of πi (xi ) and λ.
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where p is the probability that xa is the minimum. Then for any p > 0 there exists x∗ such that πie (x∗ ) > πie (xa ), which is a contradiction. For the main result, it is sufficient to show that f(¯x) diverges in the limit. When a > 0, in a minimum game, integrating f (x) = λf(x)Dx π(x) from 0 to x yields
x
f(x) = f(0) + λa
x y
(1 − G1:n−1 (y))f(y) dy − 2λb
0
0
G1:n−1 (z) dz f (y) dy
0
λa x n ( )(1 − F(y))n−2 f(y) dy n 0 1 x x − 2λb G1:n−1 (y) dyF(x) − G1:n−1 (y)F(y) dy
= f(0) +
0
λa = f(0) + G1:n (x) − 2λb n
0
x
G1:n−1 (y)[F(x) − F(y)] dy.
0
Since F ∈ [0, 1], (1 − F(x))n ≥ (1 − nF(x)). Therefore, G1:n−1 (x) ≤ nF(x) and
a f(¯x) ≥ f(0) + λ − 2bn n
x¯
F(y)(1 − F(y)) dy .
0
Suppose f(x) converges to a point-mass at x∗ < x¯ . Since π (x) < 0, by Proposition 2, for every ε > 0 and x, there exists a λε such that F(x) < ε for x ∈ [0, max(0, x∗ − ε)] and 1 − F(x) < ε for x ∈ [min(x∗ + ε, x¯ ), x¯ ] for every λ > λε . Then,
a − 2bn n
f(¯x) > λε
+
x¯ x∗ +ε
a
x∗ −ε
F(y)(1 − F(y)) dy +
0
x∗ +ε x∗ −ε
F(y)(1 − F(y)) dy
F(y)(1 − F(y)) dy
− 2bn[(x∗ − ε)ε + 2ε + (¯x − x∗ − ε)ε] n
a > λε − 2bn(¯x + 2)ε . n
> λε
With ε < (a/2bn2 (¯x + 2)), f(¯x) diverges as λ goes to infinity. Similarly, in the games with 2 ≤ j ≤ n − 1, we have f(x) = f(0) + 2λb E(mj−1:n−1 )F(x) − + 0
x y
+ λa 0
0 x
x
yf (y) dy
0
(Gj−1:n−1 (z) − Gj:n−1 (z)) dz f (y) dy
(Gj−1:n−1 (y) − Gj:n−1 (y))f(y) dy.
(A.1)
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Since
xy 0
0
425
(Gj−1:n−1 (z) − Gj:n−1 (z)) dz f(y) dy is non-negative for all λ and x,
f(¯x) ≥ f(0) + 2λb(E(mj−1:n−1 ) − E(x)) + λa
x¯ 0
(Gj−1:n−1 (y) − Gj:n−1 (y))f(y) dy
= f(0) + 2λb(E(mj−1:n−1 ) − E(x)) x¯ j−1 (n−1 [(1 − F(y))]n−j f(y) dy + λa j−1 )[F(y)] 0
λa = f(0) + 2λb(E(mj−1:n−1 ) − E(x))+ Gj:n (¯x) n a = f(0) + λ 2b(E(mj−1:n−1 ) − E(x)) + . n
(A.2)
Since logit equilibrium converges to a pure-strategy Nash equilibrium, |E(mj−1:n−1 ) − E(x)| → 0 as λ → ∞. Since a/n is independent of λ, the limiting logit equilibrium is the most efficient Nash equilibrium. References Anderson, S., Goeree, J., Holt, C., 1998. Rent Seeking with bounded rationality: an analysis of the all-pay auction. Journal of Political Economy 106, 828–853. Anderson, S., Goeree, J., Holt, C., 2001. Minimum-effort coordination games: stochastic potential and logit equilibrium. Games and Economic Behavior 34, 177–199. Bryant, J., 1983. A simple rational expectations Keynesian-type model. Quarterly Journal of Economics 98, 525– 528. Camerer, C., Ho, T.H., Chong J.-K., 2002. Sophisticated EWA learning and strategic teaching in repeated games. Journal of Economic Theory, in press. Chen, H.-C., Friedman, J., Thisse, J.-F., 1997. Boundedly Rational Nash equilibrium: a probabilistic choice approach. Games and Economic Behavior 18, 32–54. Cooper, R., John, A., 1988. Coordinating coordination failure in Keynesian models. Quarterly Journal of Economics 103, 441–463. Crawford, V.P., 1991. An ‘evolutionary’ interpretation of Van Huyck, Battalio and Beil’s experimental results on coordination. Games and Economic Behavior 3, 25–59. Crawford, V.P., 1995. Adaptive dynamics in coordination games. Econometrica 63, 103–143. McKelvey, R., Palfrey, T., 1995. Quantal response equilibria for normal-form games. Games and Economic Behavior 10, 6–38. McKelvey, R., Palfrey, T., 1998. Quantal response equilibria for extensive-form games. Experimental Economics 1, 9–41. Van Huyck, J., Battalio, R., Beil, R., 1990. Tacit coordination games, strategic uncertainty, and coordination failure. American Economic Review 80, 234–248. Van Huyck, J., Battalio, R., Beil, R., 1991. Strategic uncertainty, equilibrium selection principles, and coordination failure in average opinion games. Quarterly Journal of Economics 106, 885–910. Van Huyck, J., Battalio, R., Rankin, F., 1996. Evidence on learning in coordination games. Manuscript, Texas A&M University. Yi, K.-O., 1999. A Quantal Response Equilibrium Model of order-statistic Games. Discussion Paper 99-17, University of California, San Diego. http://www.econ.ucsd.edu/papers/files/ucsd9917.pdf.
A quantal response equilibrium model of order-statistic games Kang-Oh Yi* Department of Economics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Received 24 August 2001; accepted 15 November 2001
Abstract This paper applies McKelvey and Palfrey’s [Games Econ. Behav. 10 (1995) 6] notion of “quantal response equilibrium” to a class of “order-statistic” coordination games closely related to those studied by Van Huyck et al. [Am. Econ. Rev. 80 (1990) 234; Quart. J. Econ. 106 (1991) 885] and Van Huyck et al. [Evidence on Learning in Coordination Games. Manuscript, Texas A&M University, 1996] and those recently analyzed by Anderson et al. [Games Econ. Behav. 34 (2001) 177]. With quadratic effort costs, the limiting QRE as the noise goes to zero in their games is the most efficient equilibrium. This result contrasts with Anderson et al.’s conclusion for order-statistic games with linear effort costs, and allows a fuller assessment of the QRE’s success in describing the limiting outcomes in Van Huyck et al.’s experiments. © 2002 Elsevier Science B.V. All rights reserved. JEL classification: C79; C92 Keywords: Coordination game; Equilibrium selection; Quantal response equilibrium
1. Introduction McKelvey and Palfrey’s (1995) notion of “quantal response equilibrium” (QRE) was proposed as a way to allow realistic levels of uncertainty about players’ strategies in noncooperative game-theoretic analyses, while retaining most of the parsimony of standard equilibrium analyses. In a QRE, players do not always choose the strategy with the highest expected payoff, as in standard analyses. Instead their strategy choices are noisy, and strategies with higher payoffs are chosen with higher probabilities. As in standard equilibrium analyses, players take the noise in each other’s strategies rationally into account, correctly predicting the distributions of others’ strategies in evaluating the expected payoffs of their own strategies. ∗ Tel.: +852-2358-7619; fax: +852-2358-2084. E-mail address: [email protected] (K.-O. Yi).
0167-2681/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 6 8 1 ( 0 2 ) 0 0 0 9 5 - 1
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In applications of QRE, the noise in players’ strategy choices follows a specific distribution—often the logit distribution familiar from analyses of individual decisions—which allows the degree of noisiness to be represented by as few as one parameter. A QRE with a logit response function is called a logit equilibrium. Logit equilibrium gives a parsimonious description of imperfectly optimizing strategic behavior, which is well suited to the econometric analysis of experimental data. In some applications the noise parameter is estimated and the resulting logit equilibrium is compared with subjects’ observed choices period by period. In others, a limiting logit equilibrium, the limit of logit equilibrium as the noise approaches zero, which is usually an equilibrium in the game without noise, is compared with limiting behavior in the experiment. McKelvey and Palfrey (1995, 1998) and others have shown that the logit equilibrium is surprisingly successful in describing the quantitative as well as qualitative patterns of deviation from equilibrium observed in a variety of game experiments. Logit equilibrium also frequently implies particular patterns of selection among multiple equilibria, suggesting that it may be useful in describing the outcomes of coordination experiments. These studies have established the logit equilibrium as a standard tool in applied game theory and reserved a place for it on the list of potentially useful models of equilibrium selection. Although many qualitative properties of QRE have been established by general theoretical arguments (see for example Chen et al. (1997) and Anderson et al. (1998)), it is difficult to derive its quantitative implications analytically because it is defined as a fixed point in the space of distributions of players’ strategies. As a result, most quantitative applications of QRE have relied on numerical analyses. For similar reasons, little is known about the implications of QRE in coordination games. A notable exception on both counts is Anderson et al. (2001) (AGH), who have recently given a quantitative analytical characterization of QRE for an important class of “order-statistic” coordination games with Pareto-ranked equilibria. The games AGH studied and the games studied in this paper are closely related to those used in the classic experiments of Van Huyck et al. (1990, 1991) (VHBB) and Van Huyck et al. (1996) (VHBB). To understand AGH’s results and the results of the present paper, it is necessary to describe VHBB’s and VHBR’s games and results in more detail.1 In VHBB’s and VHBR’s games, each of n players chooses simultaneously among pure strategies called “efforts,” and players’ payoffs and best responses are determined by their own efforts and an order-statistic of their own and other players’ efforts. In VHBB (1990) the order-statistic was either the minimum of the efforts of the whole group of 14–16 (treatment A) or of pairs randomly selected from the whole group (treatment Cd ), and there was a linear payoff penalty for efforts above the minimum. In a variant of the “large-group” treatment A, the cost of effort was lowered to 0, making playing for the efficient equilibrium a weakly dominant strategy (treatment B). In VHBB (1991) the order-statistic was the median of the whole group of nine, and in the leading such treatment (Γ ) there was a quadratic penalty for efforts different from the median, which I will refer to as “quadratic effort costs” (abusing terminology somewhat because the penalty is symmetric for positive and negative deviations from the ideal effort). In VHBR’s experiments, either five or seven subjects chose among 101 efforts, the order-statistic being either the second or the fourth from the lowest effort among 1
My description follows Crawford (1991, 1995), which provides more detail on VHBB’s design and results.
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five or seven effort choices, and there was again a quadratic penalty for efforts different from the order-statistic. In each case, any symmetric effort combination is an equilibrium, and these equilibria are Pareto-ranked; but there is a tension between the higher payoffs of the Pareto-efficient equilibrium and its greater fragility, which makes it riskier to play for the efficient equilibrium when others’ responses are not perfectly predictable. These games capture important aspects of coordination problems in economic environments and resemble a number of economic models, including the models of Keynesian coordination failure in Bryant (1983) and Cooper and John (1988). Although VHBB’s and VHBR’s games all have similar structures from a game-theoretic point of view, the dynamics and limiting outcomes varied across treatments in ways that shed light on how the strategic setting affects the efficiency of coordination and that discriminate sharply among alternative models of strategic behavior. In VHBB’s (1990) treatment A, subjects converged rapidly to the least efficient equilibrium; while in treatment B, subjects converged rapidly to the most efficient equilibrium; and in treatment Cd , the outcomes were moderately inefficient but converged very slowly. In VHBB’s (1991) large-group median treatments, by contrast, subjects always locked in on the equilibrium determined by the initial median of their efforts, even though it varied across runs and was usually inefficient. In VHBR’s treatments, the results were highly heterogeneous across runs even within a treatment, sometimes converging to the efficient equilibrium and sometimes not. Thus, the limiting outcomes in VHBB’s experiments varied systematically with the number of subjects, their interaction pattern, the order-statistic, the penalty function, and the coarseness of the strategy spaces in ways that provide an interesting question for analysis. AGH’s analysis, like the present paper’s, restricts attention to games with a bounded continuum of efforts, approximating VHBR’s 101-effort games but with strategy spaces significantly less coarse than in VHBB’s original seven-effort games. AGH showed, for minimum games with linear effort costs and payoff parameter values like VHBB’s, that the limiting logit equilibrium always selects a unique equilibrium in the game without noise. This equilibrium is the least efficient equilibrium in VHBB’s treatment A, the most efficient equilibrium in treatment B, and the equilibrium with one-half of the maximum possible effort in treatment Cd . The limiting logit equilibrium therefore varies across VHBB’s (1990) minimum treatments in ways that allow it to describe the limiting experimental outcomes with surprising accuracy. AGH’s results suggest that logit equilibrium model might also be useful in describing the limiting outcomes of VHBB’s (1991) and VHBR’s experiments. VHBB’s (1991) games differ from those studied by AGH in having coarser strategy spaces (with seven efforts as in VHBB (1990)), order-statistics different than the minimum, and quadratic rather than linear effort costs. VHBR’s games have strategy spaces that approximate the bounded continuum of AGH’s model, but differ from AGH’s games in the last two ways. AGH’s analysis leaves open the question of whether the limiting logit equilibrium can also describe the very different limiting outcomes in these experiments, and if so, which of the differences just mentioned accounts for the different limiting outcomes. This paper supplements AGH’s results by providing a general, quantitative characterization of the limiting logit equilibrium in another important subclass of order-statistic games with a bounded continuum of efforts, those with quadratic effort costs, as in VHBB’s (1991)
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and VHBB’s (1996) games.2 The main result is that, for all such games, no matter what the order-statistic is, the limiting logit equilibrium is the Pareto-efficient equilibrium. Thus, AGH’s result for large-group minimum games with linear effort costs is reversed for quadratic effort costs. The analysis shows that this is a natural consequence of the near-neutrality of the logit equilibrium in order-statistic games with a continuum of efforts and the “small” (locally negligible) cost of deviating from an equilibrium with quadratic effort costs. This characterization makes it possible to compare the limiting logit equilibrium with a much more varied set of experimental outcomes. Overall, my results suggest that the limiting logit equilibrium is significantly less successful in describing the limiting outcomes of VHBB’s (1991) and VHBB’s (1996) experiments with median and other order-statistic games with quadratic effort costs than in VHBB’s (1990) minimum games with linear effort costs.3 My results also illustrate the delicacy of logit equilibrium analysis in coordination games with fine strategy spaces, highlighting the importance of seemingly minor details of the cost function. I close by briefly discussing subjects’ incentives, in games with fine strategy spaces and quadratic effort costs, to use small variations in an effort to signal a willingness to move to a more efficient equilibrium. This suggests a partial explanation of the otherwise puzzling outcomes of VHBR’s experiment, in the spirit of Camerer et al. (2002) analysis of “strategic teaching.” The rest of the paper is organized as follows. Section 2 introduces a class of order-statistic games with bounded, continuous strategy spaces and characterizes their logit and limiting logit equilibria. Section 3 studies subjects’ incentives to signal in games with fine strategy spaces and quadratic effort costs, suggesting a partial explanation of VHBR’s results. Section 4 is the conclusion. Proofs omitted from the text are in Appendix A.
2. Order-statistic games and logit equilibrium In an n-person jth order-statistic game, each player chooses an effort level xi ∈ [0, x¯ ], i = 1, . . . , n, where x¯ is a finite maximum effort level. Each player is assumed to be risk-neutral. Let ui (xi , mj:n ) be the player i’s payoff when he plays xi and the resulting order-statistic is mj:n . In this paper, I use a specific functional form which has been used in Van Huyck et al. (1991, 1996). ui (xi , mj:n ) = amj:n − b(mj:n − xi )2 + c,
a, b, c ≥ 0,
(1)
where mj:n is the jth inclusive order-statistic which is defined by m1:n ≤ m2:n ≤ · · · ≤ mn:n , where the mj:n is the jth element of choice combinations {x1 , . . . , xn } arranged in increasing order. In QRE, the probability density of player i’s choosing xi is a function of the expected payoff πie (xi ) and the density of each choice is an increasing function of the expected payoff 2 My research for the first version of this paper (Yi, 1999) was independent of AGH’s work, although the current version has benefited significantly from their analysis. 3 Crawford (1995) proposes an alternative explanation of Van Huyck et al.’s (1990, 1991) results, which also misses some important features of VHBR’s results.
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for that choice. In this paper, I focus on a specialized version of QRE where the choice probabilities are an analogue of the standard multinomial logit distribution as in AGH. fi (xi ) = x¯ 0
exp(λπie (xi ))
exp(λπie (y)) dy
,
(2)
where 0 ≤ λ < ∞ measures the amount of noise, or equivalently, the degree of rationality. This functional form is called a logit function where the odds are determined by the exponential transformation of the utility times a given non-negative constant λ.4 As λ → ∞, the probability of the choice having the highest expected payoff becomes 1, if it is unique, so that the choice behavior becomes best response; when λ = 0 all choices have equal probability. Logit equilibrium for a given λ is defined by a fixed point in these probability distributions. In other words, in equilibrium players’ logit response functions are mutually “noisy best responses,” given λ. To apply the probabilistic choice rule, Eq. (2), to order-statistic games, we need to compute expected payoff, πie (xi ). Let Fi (x) denote the cumulative distribution function associated with fi (x). Let Gij:n−1 (x) be the cumulative distribution function of jth order-statistic regarding {x1 , . . . , xi−1 , xi+1 , . . . , xn } where the xs are drawn from distributions, F−i = {F1 , . . . , Fi−1 , Fi+1 , . . . , Fn }, respectively. That is, Gij:n−1 (x) =
n−1 k k=j Sk l=1
Fil (x)
n−1
(1 − Fil (x)),
l=k+1
where the summation Sk extends over all permutations i1 , . . . , in−1 of 1, . . . , n − 1 for which i1 < · · · < ik and ik+1 < · · · <in−1 . When j = 1, this can be simplified as Gi1:n−1 (x) = 1 − i=k (1 − Fi (x)) where i=k Fi (x) = F1 (x) · · · Fk−1 (x) · · · Fk+1 (x) · · · i (x) be the associated probability density function. Since the logit effort Fn (x). Let gj:n−1 density is invariant to the changes in the origin, the expected payoff can be normalized by excluding those terms that are independent of xi . The normalized expected payoffs in order-statistic games are given in Lemma 1. Lemma 1. In an n-person minimum game, the part of expected payoff that is relevant to logit equilibrium (the part that depends on xi ) is xi xi i i πi (xi ) = a (1 − G1:n−1 (y)) dy + 2b (y − xi )G1:n−1 (y) dy . (3) 0
0
In an n-person order-statistic game with 2 ≤ j ≤ n − 1, the part of expected payoff that is relevant to logit equilibrium is πi (xi ) = 2bEi (mj−1:n−1 )xi − bx2i xi + (a − 2by + 2bxi )(Gij−1:n−1 (y) − Gij:n−1 (y)) dy.
(4)
0
4 In principle, logit equilibrium permits players to have different λs, but the common knowledge assumption is indispensible for the present analysis.
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Substituting πi (xi ) into Eq. (2) completes the determination of the effort densities in order-statistic games. For finite normal-form games, McKelvey and Palfrey (1995) proved the existence and the convergence of logit equilibrium to a Nash equilibrium as λ goes to infinity. In order-statistic games, the same properties hold with continuous strategy spaces. Proposition 1. In any order-statistic game with quadratic effort costs, there exists a logit equilibrium for every λ ≥ 0. The proof of Proposition 1 is a simple modification of the existence proof in Anderson et al. (1998). For a complete proof, see Yi (1999). Given the existence of logit equilibrium, since infinite λ is the “rationality” limit, a limiting logit equilibrium gives a selection from the set of Nash equilibria (McKelvey and Palfrey, 1995). In general, logit equilibrium and limiting logit equilibrium are not unique. In particular, in finite games with multiple strict equilibria, any strict Nash equilibrium can be found as a limit of logit equilibrium.5 But Proposition 2 shows that the limiting logit equilibrium is unique in the order-statistic games studied here. Proposition 2. In any order-statistic game with quadratic effort costs, as λ goes to infinity, the logit equilibrium converges to the most efficient Nash equilibrium, xi = x¯ for all i. Proposition 2 shows that, with quadratic effort costs, as long as a > 0 the limiting logit equilibrium is always efficient regardless of the order-statistic, the number of players, and the values of payoff parameters. The intuition behind this result is that, at xi∗ = Ei (mj:n |xi∗ ), the expected benefit to an increase in one’s own effort is linear but the cost is quadratic, so for a sufficiently small increase in one’s effort the benefit always exceeds the cost. This small “tilt” in favor of higher efforts allows the limiting logit equilibrium with quadratic effort costs to be very different than the limiting logit equilibrium with linear effort costs, even when the order-statistic is the minimum. To see the relation between the benefit and cost of increasing effort more clearly, consider the following payoff function. ui (xi , mj:n ) = amj:n − b(mj:n − xi )2 + c − dxi ,
d ≥ 0.
The associated expected payoff is πie (xi ) = aEi (mj:n |xi ) − bEi [(mj:n |xi ) − Ei (mj:n |xi )]2 + c − dxi = aEi (mj:n |xi ) − bVar i (mj:n |xi ) + c − dxi , where Var(·) denotes the variance. Its first derivative with respect to xi is ∂πie (xi ) ∂Ei (mj:n |xi ) ∂Var i (mj:n |xi ) =a −b − d. ∂xi ∂xi ∂xi
(5)
5 By definition, a strict equilibrium is necessarily a pure-strategy equilibrium. Strict equilibria remain strict when the payoffs are slightly perturbed.
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Given other players’ strategies, the first term on the right-hand side is strictly positive as long as λ is finite and the second term is zero at xi∗ , where the variance attains its minimum. Therefore, the value of a(∂Ei (mj:n |xi∗ )/∂xi ) − d indicates the direction of the best response. When d = 0, the best response effort level is always higher than xi∗ , and there is a force that pushes the effort density towards to the maximum effort level. An interior equilibrium effort level is possible only when the values of a and d preserve a delicate balance between the marginal benefit and cost. For generic values of a and d, a limiting logit equilibrium effort level will be at one or the other extreme. Note that this argument depends only on the quadratic effort costs, not on the details of the noise structure. Thus noise-based notions of equilibrium in order-statistic games with “locally small” effort costs can be expected to tip the balance of the benefit and the cost in favor of a more efficient equilibrium more generally. When a = 0, the expected payoff is πie (xi ) = −bEi (mj:n − xi )2 + c. In this case, the expected payoff is maximized at xi∗ = E(mj:n |xi∗ ). As a result, the limiting logit equilibrium depends on the statistical properties of the order-statistic via j and n. For instance, in a minimum game, since effort level 0 is a unique weakly dominant strategy and there is no mixed-strategy Nash equilibrium, the limiting logit equilibrium should be xi = 0 for all i as λ goes to infinity. The proof for other order-statistic games with a = 0 is quite involved and tedious, and is omitted. See Yi (1999) for the complete proof. Proposition 3. In an order-statistic game with a quadratic payoff function with a = 0, the limiting logit equilibrium for j < (n/2) is xi = 0 for all i; and for j > (n/2) + 1, it is xi = x¯ for all i.
3. Logit equilibrium and experimental results Although the results in AGH and the previous section show that the limiting logit equilibrium gives a unique prediction in order-statistic games, it fails to match some of the experimental results. As AGH showed, VHBB’s (1990) experimental results for minimum games with linear effort costs correspond closely to the limiting logit equilibrium in each of VHBB’s treatments A, B, and Cd . However, VHBB’s (1991) and VHBB’s (1996) experimental results differ from the efficient limiting logit equilibrium identified in Proposition 2. This section explains in more detail how Proposition 2 relates to VHBB’s (1991) and VHBB’s (1996) results and how the coarseness of the strategy space affects players’ incentives. In VHBB’s (1991) experiment, nine subjects played a median game with Xi = {1, 2, . . . , 7} and d = 0. In treatment Γ , a = 0.10 and b = 0.05, and treatment Φ used a = 0 and b = 0.05.6 In both treatments, subjects’ efforts invariably converged by the last period to the initial median, which was effort 4 in some runs and effort 5 in others. 6 There was also a treatment, Ω, where a player’s payoff increases linearly in the median effort, but the payoff is 0 when his effort is different from the order-statistic. Since the payoff structure is very different from one analyzed here, treatment Ω is not considered.
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In VHBR’s experiment, either five or seven subjects played a second- or fourth-orderstatistic game with 101 effort levels, and with the identical payoff parameters used in treatment Γ in VHBB (1991). The strategy space was ei ∈ {0, 1, . . . , 100}, and the map used to determine payoffs was xi = 1 + 0.06ei . Such a transformation brings the environment much closer to the continuum of efforts of AGH’s and my models. It also led to results that differed from VHBB’s (1991) in surprising ways. In VHBR’s experiment, the play converged to the efficient equilibrium in 15 out of 36 runs, with significantly less history dependence than appeared in VHBB (1991). The dynamics suggest that may subjects focused on the current order-statistic and then engaged in local experimentation that are skewed upward, in the direction of efficiency.7 The fact that this experimentation occurred in VHBR’s but not VHBB’s (1991) experiment may be due to the effect of VHBR’s finer strategy space on the cost of small experimentation. Letting mj:n (x1 , . . . , xn ) be an order-statistic and ei = α + βxi with β ≥ 0, we have mj:n (e1 , . . . , en ) = α + βmj:n (x1 , . . . , xn ). From Eq. (5), transforming the strategy space from xi to ei gives %πie (ei ) %Ei (mj:n | ei ) %Var i (mj:n | ei ) =β a − βb −d . (6) %ei %ei %ei When player i raises the effort by one unit, on average, the relative effort cost to the benefit is reduced by 16 times more in VHBB’s (1996) games than in VHBB’s (1991) game. Given this observation, it is natural to ask why AGH’s analysis of limiting logit equilibrium with a continuum of efforts was so successful in predicting the limiting outcomes in VHBB’s (1990) treatments with coarse effort spaces. Recall that VHBB’s (1990) minimum game experiment used essentially three different treatments A, B, and Cd with Xi = {1, 2, . . . , 7}. Treatments A and B are played by 14-16 players, and two players, randomly paired with new partners each period, played each game in treatment Cd .8 In treatments A and Cd , a = 0.20, b = 0, and d = 0.10, while in treatment B, d = 0. This difference may be due to the fact that, since b = 0 for all treatments, AGH’s continuous approximation to the seven-effort games barely affects players’ cost of experimentation in contrast to the strong effect in Eq. (6). In each treatment, the dynamics yield a limiting outcome surprisingly close to the limiting logit equilibrium that is xi = 0 if d > a/n; xi = (¯x/n) if d = (a/n); and xi = x¯ if d < (a/n) for all i (AGH). This difference may explain some, but not all, of the difference between VHBB’s (1991) and VHBB’s (1996) results. It appears that a model in which players focus on stagegame strategies is not enough for a full explanation. Along this line of research, there are some potential explanations connecting history dependence and grid size. VHBR suggested that a fine action grid lowers the opportunity cost of exploring risky but mutually 7 The average change in effort choices is small ranging from −0.54 to 2.62, and in all but one run the value is positive. 8 In the random-pairing treatment C each subject was told only his current pair’s minimum. There was also d a treatment, Cf , in which the minimum game was repeatedly played by fixed pairs, with subjects informed that their pairing were fixed. As Crawford (1995) notes, those subjects were evidently aware of the importance of repeated-game strategies and usually achieved efficient coordination. Because repeated-game strategies raise difficult new issues, that treatment is not discussed here.
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beneficial strategies and thus facilitates coordination on a better outcome. Camerer et al. (2002) proposed a model of “strategic teaching,” in which players choose strategies with poor short-run payoffs expecting better long-run payoffs when there are enough “sophisticated” players, which might provide a way to formalize VHBR’s suggested explanation.
4. Concluding remarks This paper provides a general characterization of logit equilibrium in a class of orderstatistic games with a bounded continuum of efforts and quadratic effort costs, supplementing AGH’s (2001) analysis of similar games with linear effort costs. The results show that the limiting logit equilibrium is significantly less successful in describing VHBB’s (1991) and VHBB’s (1996) results than VHBB’s (1990) results. They also highlight the subtlety of players’ incentive problems in order-statistic games with respect to the coarseness of strategy spaces and the functional form of cost of effort.
Acknowledgements I am deeply grateful to Vincent Crawford and Joel Sobel for their advice and encouragement.
Appendix A Proof of Lemma 1. In a minimum game, xi i πie (xi ) = (ay − b(y − xi )2 )g1:n−1 (y) dy + axi (1 − Gi1:n−1 (xi )) + c 0 xi xi Gi1:n−1 (y) dy + 2b yGi1:n−1 (y) dy = axi Gi1:n−1 (xi ) − a 0 0 xi − 2bxi Gi1:n−1 (y) dy + axi − axi Gi1:n−1 (xi ) + c. 0
Since the logit effort density is invariant to the changing of the origin, c is irrelevant in determining the equilibrium density function. When 2 ≤ j ≤ n − 1, given xi , P(mj:n < y) = P(mj:n−1 < y) = Gij:n−1 (y),
for y < xi
P(mj:n < y) = P(mj−1:n−1 < y) = Gij−1:n−1 (y),
for y > xi
P(mj:n = y) = P(mj−1:n−1 ≤ y ≤ mj:n−1 ) = Gij−1:n−1 (y) − Gij:n−1 (y),
for y = xi .
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Then the expected payoff becomes xi ygij:n−1 (y) dy − b πie (xi ) = a 0
x¯
0
i (y − xi )2 gj:n−1 (y) dy
x¯
x¯
i (y − xi )2 gj−1:n−1 (y) dy xi xi xi i i i +ax(Gj−1:n−1 (xi ) − Gj:n−1 (xi )) + c = −a y(gj−1:n−1 (y) 0 xi i i i (y)) dy + b (y − xi )2 (gj−1:n−1 (y) − gj:n−1 (y)) dy −gj:n−1
+a
+a 0
ygij−1:n−1 (y) dy
xi
−b
0
x¯
ygij−1:n−1 (y) dy − b
0
i (y − xi )2 gj−1:n−1 (y) dy
+ax(Gij−1:n−1 (xi ) − Gij:n−1 (xi )) + c = −axi (Gij−1:n−1 (xi ) xi i −Gj:n−1 (xi )) + a (Gij−1:n−1 (y) − Gij:n−1 (y)) dy + bx2i (Gij−1:n−1 (xi ) 0 xi y(Gij−1:n−1 (y) − Gij:n−1 (y)) dy −Gij:n−1 (xi )) − 2b 0 xi 2 i (Gij−1:n−1 (y) −2bxi (Gj−1:n−1 (xi ) − Gij:n−1 (xi )) + 2bxi
0 i 2 i i −Gj:n−1 (y)) dy + bxi (Gj−1:n−1 (xi ) − Gj:n−1 (xi )) + aEi (mj−1:n−1 ) −bEi (m2j−1:n−1 ) + 2bxi Ei (mj−1:n−1 ) − bx2i + axi (Gij−1:n−1 (xi ) −Gij:n−1 (xi )) + c = aEi (mj−1:n−1 ) − bEi (m2j−1:n−1 ) + 2bEi (mj−1:n−1 )xi xi −bx2i + (a − 2by + 2bxi )(Gij−1:n−1 (y) − Gij:n−1 (y)) dy + c. 0
Since the logit effort density is invariant to the changing of the origin, the first two terms in the last equation are irrelevant in determining the equilibrium density function as well as c. Proof of Proposition 2. Given existence, Theorem 2 and Lemma 2 in McKelvey and Palfrey (1995) show that any limiting logit equilibrium is a Nash equilibrium. Having the convergence result, the proof consists of two parts. After I show that any logit equilibrium is symmetric across players and every symmetric Nash equilibrium is a pure-strategy equilibrium, I show that logit equilibrium selects the Pareto-efficient equilibrium. The first part of the proof concerns only a minimum game. Basically the similar argument applies to the other order-statistic games, and for a complete proof see Yi (1999). To show symmetry by a contradiction, suppose that player 1 and 2have different logit equilibrium densities. Although they have different effort densities, they should hold the beliefs about the other (n − 2) players’ strategies. Let Qj:n−2 (x) be the probability that exactly j of the xi s are less than or equal to x and n − j − 2 are greater than or equal to x in n − 2 random
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samples drawn from {F3 , . . . , Fn }. Then, G1j:n−1 (x) = Qj−1:n−2 (x)F2 (x), and G2j:n−1 (x) = Qj−1:n−2 (x)F1 (x), where Q0:n−2 (x) =
n−2
(1 − Fi (x)).
i=3
Without loss of generality, since f1 (x) and f2 (x) should cross at least once, f1 (x) > f2 (x) and F1 (x) > F2 (x) on (xa , xb ) and f1 (xb ) = f2 (xb ), where xa is the smallest one from which two densities have different values and xa could be 0. Let H(x) ≡ (1/λ)((f1 (x)/f1 (x)) − (f2 (x)/f2 (x))) = Dx π1 (x) − Dx π2 (x).9 Then we have H(xb ) < 0, or xb G21:n−1 (y) − G11:n−1 (y) dy H(xb ) = a(G21:n−1 (xb ) − G11:n−1 (xb )) + 2b 0 xb (F1 (y)−F2 (y))Q0:n−2 (y) dy < 0. = a(F1 (xb ) − F2 (xb ))Q0:n−2 (x) + 2b 0
However, this contradicts F1 (x) > F2 (x) for all x ∈ {xa , xb }. To show that any symmetric equilibrium is pure by a contradiction, suppose that there is a mixed-strategy Nash equilibrium that could have three forms, either the effort levels played with positive probability are isolated or the support involves contiguous intervals or both. Consider the possibility of a contiguous interval first. Let xm and xm be two effort levels in the same interval such that xm < xm . If the support of an equilibrium strategy involves an interval [xm , xm ], then a player’s expected payoff must be constant over the interval and the first derivative of expected payoff exists and must be zero as well as the second derivative. For x ∈ [xm , xm ], however, Dx2 πe (x) = −(a + 2b)g1:n−1 (x) < 0, which is a contradiction. Hence, the equilibrium density can involve only atoms. Now suppose that more than one effort level are played with a positive probability. Let xa denote the minimum effort level and let xb be the second largest effort level being played with a positive probability. Then the expected payoff from xa is πie (xa ) = axa + c, which should be greater than or equal to the expected payoffs for all efforts x∗ ∈ (xa , xb ), πie (x∗ ) = p[axa − b(xa − x∗ )2 ] + (1 − p)ax∗
= axa − bp(xa − x∗ )2 + a(1 − p)(x∗ − xa ),
9 Logit equilibrium effort density f (x ) is jointly continuous in x and λ, and differentiable with respect to x . i i i i It is because πi (xi ) is jointly continuous in xi and λ, and differentiable with respect to xi , and fi (xi ) is the ratio of continuous transformation of πi (xi ) and λ.
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where p is the probability that xa is the minimum. Then for any p > 0 there exists x∗ such that πie (x∗ ) > πie (xa ), which is a contradiction. For the main result, it is sufficient to show that f(¯x) diverges in the limit. When a > 0, in a minimum game, integrating f (x) = λf(x)Dx π(x) from 0 to x yields
x
f(x) = f(0) + λa
x y
(1 − G1:n−1 (y))f(y) dy − 2λb
0
0
G1:n−1 (z) dz f (y) dy
0
λa x n ( )(1 − F(y))n−2 f(y) dy n 0 1 x x − 2λb G1:n−1 (y) dyF(x) − G1:n−1 (y)F(y) dy
= f(0) +
0
λa = f(0) + G1:n (x) − 2λb n
0
x
G1:n−1 (y)[F(x) − F(y)] dy.
0
Since F ∈ [0, 1], (1 − F(x))n ≥ (1 − nF(x)). Therefore, G1:n−1 (x) ≤ nF(x) and
a f(¯x) ≥ f(0) + λ − 2bn n
x¯
F(y)(1 − F(y)) dy .
0
Suppose f(x) converges to a point-mass at x∗ < x¯ . Since π (x) < 0, by Proposition 2, for every ε > 0 and x, there exists a λε such that F(x) < ε for x ∈ [0, max(0, x∗ − ε)] and 1 − F(x) < ε for x ∈ [min(x∗ + ε, x¯ ), x¯ ] for every λ > λε . Then,
a − 2bn n
f(¯x) > λε
+
x¯ x∗ +ε
a
x∗ −ε
F(y)(1 − F(y)) dy +
0
x∗ +ε x∗ −ε
F(y)(1 − F(y)) dy
F(y)(1 − F(y)) dy
− 2bn[(x∗ − ε)ε + 2ε + (¯x − x∗ − ε)ε] n
a > λε − 2bn(¯x + 2)ε . n
> λε
With ε < (a/2bn2 (¯x + 2)), f(¯x) diverges as λ goes to infinity. Similarly, in the games with 2 ≤ j ≤ n − 1, we have f(x) = f(0) + 2λb E(mj−1:n−1 )F(x) − + 0
x y
+ λa 0
0 x
x
yf (y) dy
0
(Gj−1:n−1 (z) − Gj:n−1 (z)) dz f (y) dy
(Gj−1:n−1 (y) − Gj:n−1 (y))f(y) dy.
(A.1)
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Since
xy 0
0
425
(Gj−1:n−1 (z) − Gj:n−1 (z)) dz f(y) dy is non-negative for all λ and x,
f(¯x) ≥ f(0) + 2λb(E(mj−1:n−1 ) − E(x)) + λa
x¯ 0
(Gj−1:n−1 (y) − Gj:n−1 (y))f(y) dy
= f(0) + 2λb(E(mj−1:n−1 ) − E(x)) x¯ j−1 (n−1 [(1 − F(y))]n−j f(y) dy + λa j−1 )[F(y)] 0
λa = f(0) + 2λb(E(mj−1:n−1 ) − E(x))+ Gj:n (¯x) n a = f(0) + λ 2b(E(mj−1:n−1 ) − E(x)) + . n
(A.2)
Since logit equilibrium converges to a pure-strategy Nash equilibrium, |E(mj−1:n−1 ) − E(x)| → 0 as λ → ∞. Since a/n is independent of λ, the limiting logit equilibrium is the most efficient Nash equilibrium. References Anderson, S., Goeree, J., Holt, C., 1998. Rent Seeking with bounded rationality: an analysis of the all-pay auction. Journal of Political Economy 106, 828–853. Anderson, S., Goeree, J., Holt, C., 2001. Minimum-effort coordination games: stochastic potential and logit equilibrium. Games and Economic Behavior 34, 177–199. Bryant, J., 1983. A simple rational expectations Keynesian-type model. Quarterly Journal of Economics 98, 525– 528. Camerer, C., Ho, T.H., Chong J.-K., 2002. Sophisticated EWA learning and strategic teaching in repeated games. Journal of Economic Theory, in press. Chen, H.-C., Friedman, J., Thisse, J.-F., 1997. Boundedly Rational Nash equilibrium: a probabilistic choice approach. Games and Economic Behavior 18, 32–54. Cooper, R., John, A., 1988. Coordinating coordination failure in Keynesian models. Quarterly Journal of Economics 103, 441–463. Crawford, V.P., 1991. An ‘evolutionary’ interpretation of Van Huyck, Battalio and Beil’s experimental results on coordination. Games and Economic Behavior 3, 25–59. Crawford, V.P., 1995. Adaptive dynamics in coordination games. Econometrica 63, 103–143. McKelvey, R., Palfrey, T., 1995. Quantal response equilibria for normal-form games. Games and Economic Behavior 10, 6–38. McKelvey, R., Palfrey, T., 1998. Quantal response equilibria for extensive-form games. Experimental Economics 1, 9–41. Van Huyck, J., Battalio, R., Beil, R., 1990. Tacit coordination games, strategic uncertainty, and coordination failure. American Economic Review 80, 234–248. Van Huyck, J., Battalio, R., Beil, R., 1991. Strategic uncertainty, equilibrium selection principles, and coordination failure in average opinion games. Quarterly Journal of Economics 106, 885–910. Van Huyck, J., Battalio, R., Rankin, F., 1996. Evidence on learning in coordination games. Manuscript, Texas A&M University. Yi, K.-O., 1999. A Quantal Response Equilibrium Model of order-statistic Games. Discussion Paper 99-17, University of California, San Diego. http://www.econ.ucsd.edu/papers/files/ucsd9917.pdf.