Invader strategies in the war of attrition with private information

Journal of Mathematical Economics 50 (2014) 160–166

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Invader strategies in the war of attrition with private information✩ Lars Peter Metzger ∗ Department of Economics, University of Dortmund, 44221 Dortmund, Germany

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Article history: Received 5 October 2012 Received in revised form 28 October 2013 Accepted 23 November 2013 Available online 12 December 2013 Keywords: Evolutionary stability War of attrition Strict equilibrium Neighborhood invader strategy Continuous stability Evolutionary robustness

abstract Second price all-pay auctions (wars of attritions) have an evolutionarily stable equilibrium in pure strategies if valuations are private information. I show that for any level of uncertainty there exists a pure deviation strategy arbitrarily close to the equilibrium strategy such that for some valuations the equilibrium strategy has a selective disadvantage against the deviation if the population mainly plays the deviation strategy. I show that agents with those valuations would prefer to deviate even farther from the equilibrium strategy, if the population collectively uses the deviation strategy. I argue that in the Bayesian game studied here, a mass deviation can be caused by the entry of a small group of agents. The results provided in this paper imply that the equilibrium strategy is indeed unstable if one considers rare and independent mutations on the space of valuations. Numeric calculations indicate that the closer the deviation strategy to the equilibrium strategy, the more valuations are destabilizing. © 2013 Elsevier B.V. All rights reserved.

1. Introduction and Literature In this paper I analyze the stability of equilibrium behavior in second price all-pay auctions of incomplete information with two contestants. An all-pay auction is a contest in which each contestant exerts efforts that are foregone regardless of winning the prize. In a second price all-pay auction, the winner pays the second highest bid and all other contestants pay their own bid. This contest is also known as the ‘war of attrition’ which was introduced by Maynard Smith (1974) to exemplify his concept of evolutionary stability. Beside the biological interpretation, an all-pay auction is a situation to which social agents are exposed in daily routine: a successful job market candidate needs to be better qualified than the second best candidate, a sprinter needs to poke his or her nose a fraction of a second over the finish line before the second fastest athlete. Electoral first-pass-the-post campaigns, lobbying, academic research, public invasions to tender, and irreducible investments with conditional stochastic yields are all examples in which monetary or non-monetary spendings are sunk before the final allocation of the prize is fixed. These situations also share the property that the absolute value of the bid is irrelevant—what matters is the relative bid intensity. I study contests in which the

✩ I am grateful to Carlos Alós-Ferrer, Tymon Tatur, Eugen Kóvac, Asen Kochov, Thomas Gall, Burkard Hehenkamp, Wolfgang Leininger, Frank Riedel, Klaus Ritzberger, two anonymous referees and the session auditories of ESEM 2011 in Oslo and SAET 2011 in Faro for helpful and constructive comments. ∗ Tel.: +49 2317553804. E-mail address: [email protected].

0304-4068/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmateco.2013.11.006

valuation for the prize is private information. Situations such as in Moldovanu and Sela (2001) or Ponsati and Sákovics (1995) in which the prize is equally valuable for all contestants but the cost of exerting efforts differ are every bit as plausible as the setting chosen here and can be seen as equivalent after a transformation of payoffs. Maynard Smith (1974)’s ‘war of attrition’ and related all-pay auctions have been shown to be the limit of other, more general models as in Abreu and Gul (2000) who develop a model of reputation based bargaining or Lang et al. (2010) who analyze stochastic (Poisson) contests and Che and Gale (2000) who analyze rent seeking games. Bishop et al. (1978) characterize the evolutionarily stable strategy for the case of incomplete information. Milgrom and Weber (1985) show that as uncertainty approaches zero, the distribution of the (pure) bids converges to the mixed strategy distribution of Maynard Smith (1974). The war of attrition with incomplete information has also been studied by Nalebuff and Riley (1985) who show that the game has a continuum of asymmetric equilibria if the population of agents is subdivided into observationally distinguishable but payoff irrelevant classes. In this paper we study the symmetric single population game with a unique equilibrium. It is understood that the war of attrition or other all-pay auctions can be found in many economic applications, such as the IO models ‘The Generalized War of Attrition’ in Bulow and Klemperer (1999) or Konrad (2006). Rose (1978) studies the evolutionary stability of all-pay first price auctions (Scotch Auctions). The stability of first price auctions in which only the winner pays has been studied by Hon-Snir et al. (1998) and Louge and Riedel (2010). The war of attrition in finite

L.P. Metzger / Journal of Mathematical Economics 50 (2014) 160–166

populations has been studied by Riley (1980), all-pay auctions (Tullock-contest) have been shown to exhibit non-Nash behavior for finite populations by Hehenkamp et al. (2004). Damianov et al. (2010) investigate whether a uniform or a discriminatory price auction is better for the seller in an experiment. Bishop et al. (1978) use the concept of evolutionary stability in a game with continuous strategies. For such games it has been proposed to use other concepts such as (local) neighborhood invader strategy (Apaloo, 1997, 2006), continuously stable strategy (Eshel and Motro, 1981), evolutionary robustness (Oechssler and Riedel, 2002), and asymmetric versions of continuous stability and neighborhood invader strategy (Cressman, 2010), because it has been shown that evolutionary stability is not sufficient for dynamic stability if strategies are continuous. Bishop and Cannings (1978) show convergence to the evolutionarily stable strategy in their ‘Generalized War of Attrition’ only for finite strategy sets. To stress that the critique of the use of evolutionary stability is long known I quote Hofbauer et al. (1979, p. 611), ‘‘(. . . ) [I]t could be that under certain circumstances it would be more appropriate to study asymptotically stable equilibria of (l), rather than ESS’’. Eq. (1) is the replicator dynamic (Taylor and Jonker, 1978). While I do not study asymptotic stability in Taylor and Jonker’s (1978) replicator dynamic I take up Hofbauer, Sigmund and Schuster’s (1979) critique and analyze whether the static conditions of the previous paragraph that ensure dynamic stability in the replicator dynamic are met. The effect of the discretization of a continuous game is the subject of Alós-Ferrer (2006). Also Boudreau (2010) studies all-pay auctions with discrete action spaces. Krishna and Morgan (1997) develop a model in which all-pay auctions raise more expected revenue than other sealed-bid auction forms. Leininger (2000) sees the all-pay auction as a benchmark lottery and discusses the role of the revenue equivalence theorem in understanding the differences of auction types. This paper adds to the literature that analyzes the dynamic stability of equilibrium strategies in auctions. In the current setting, a strategy is a mapping from a continuum of types (valuations) into the non-negative reals. For such strategies the literature does not agree on the notion of stability. I prove the existence of an invader strategy which is destabilizing for an open set of valuations for any continuous distribution of valuations. I show that the equilibrium strategy is not continuously stable (Eshel and Motro, 1981) and that it is no (local) neighborhood invader strategy (Apaloo, 1997, 2006). I hereby claim that the equilibrium strategies in the war of attrition with private valuations is not dynamically stable. Section 2 presents the static model and its equilibrium, Section 3 discusses the use of the stability concept. Sections 4 and 5 collect the analytic respective the numeric results and Section 6 concludes. 2. The static model Let there be two contestants, each having a valuation in the nonempty set V ⊂ R+ , where V is a closed interval. The valuations are distributed according to a cumulative distribution function F with continuous positive density f . Let B = R+ be the set of bids that a contestant can choose from. A pure bid-strategy is a mapping β : V → B that assigns for each valuation v ∈ V a bid β(v) ∈ B . If one contestant uses strategy β : V → B and the other contestant has valuation v ∈ V and chooses b ∈ B , he receives expected

161

Fig. 1. Level curves and equilibrium strategy for uniform valuations.

payoffs

  (v − β(w))f (w)dw    {w:β(w)b} bf (w)dw  does not get the prize and pays own bid. If F ({w : β(w) = b}) = 0 for all b ∈ B , the payoffs can be expressed as

π (b|v, β) =



(v + b − β(w))f (w)dw − b. {w:β(w)
For textbook discussions of the war of attrition see for example Klemperer (2004, part 2.2.2) or Fudenberg and Tirole (1991, part 4.5.2). Bishop et al. (1978) show that the unique Bayesian Nash equilibrium consists of the strategy

β(v) =

v

 0

w f (w) dw. 1 − F (w)

If F is the uniform distribution on [0, 1], then

β(v) = − ln(1 − v) − v. Fig. 1 depicts the level curves of π (b|v, β) (dashed lines) and the equilibrium strategy β (solid line) if valuations are distributed uniformly on [0, 1]. In the common value case in which each contestant has the same valuation for the price, v¯ say, Maynard Smith (1974) shows that the unique symmetric equilibrium consists of the mixed strategy σ (b) = v1¯ e−b/¯v .1 Maynard Smith (1974) shows that the mixed strategy is an evolutionarily stable strategy in the game of complete information and Bishop et al. (1978) show evolutionary stability of the pure strategy equilibrium in the game of incomplete information. Milgrom and Weber (1985) argue that if F is uniformly concentrated on a neighborhood (¯v −ϵ, v¯ +ϵ) then the equilibrium distribution of bids converges to the distribution induced by σ (b) as ϵ → 0.

1 There are asymmetric equilibria (b, d) with b ≥ v¯ and d = 0.

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L.P. Metzger / Journal of Mathematical Economics 50 (2014) 160–166

3. Dynamic stability To analyze dynamic stability, we have the following interpretation of the model: suppose that there is an infinite population of contestants each having a fixed valuation such that the distribution of valuations matches F . To play the contest, two agents are independently and uniformly matched. For each agent a strategy is an element of B rather than a mapping β : V → B . Bishop et al. (1978) show that the condition for evolutionary stability does hold for each valuation. As Oechssler and Riedel (2002) point out the standard notion of evolutionarily stable strategy (Maynard Smith, 1974) is not sufficient for dynamic stability in games of infinite strategy sets. For such games the concepts continuously stable strategy (Eshel and Motro, 1981), (local) neighborhood invader strategy (Apaloo, 1997, 2006), evolutionary robustness (Oechssler and Riedel, 2002) and neighborhood superiority (Cressman, 2005) have been proposed. Any of these concepts require stability against mass deviations. A mass deviation describes a situation in which each agent of the population simultaneously deviates to an identical strategy. This seems to be a very odd and implausible event as ‘trembles’ or ‘mutations’ are usually seen as independent events. Why can the whole population independently and undirectedly ‘mutate’ to the same deviation strategy? In the Bayesian game considered here I show that there is a correlation device that gives a plausible interpretation for mass deviations and that such a mass deviation is triggered by a deviation of an arbitrarily small subgroup of agents. 3.1. Stability concepts For symmetric games with finite sets of strategies Maynard Smith (1974) defines the concept evolutionarily stable strategy. If a homogeneous population of agents playing the evolutionarily stable strategy is invaded by a small portion of mutants, the agents who play the evolutionarily stable strategy have a selective advantage against the mixture in which the population mainly plays the evolutionarily stable strategy and a small fraction deviates to the mutant strategy. The concept technically allows for two interpretations of mutations. The first and usually preferred interpretation is that mutations happen independently and undirectedly. This means that only a very small fraction of agents mutates and that the mutant strategy can be any strategy of the set of strategies. The second interpretation, merely from a correlated shocks perspective, is that almost all agents change their strategy but the new strategy must be very similar to the original one. Hofbauer et al. (1979) show that a strategy is an evolutionarily stable strategy if and only if it satisfies π (x, y) > π (y, y) for all mixed strategies y ̸= x, y close to x. π (x, y) is the payoff to an agent who uses strategy x against an opponent who uses strategy y. As in this context the game is finite, close is to be understood in the usual meaning of the Euclidean distance. Weibull (1995) coins this condition local superiority. It is exactly this condition which was used by Hofbauer et al. (1979) to construct a Lyapunov function to show asymptotic stability in the replicator dynamic. The stability concepts that follow all consider games with continuous sets of strategies. The neighborhood invader strategy (Apaloo, 2006) demands that if the population mainly uses a strategy y that is distinct to the neighborhood invader strategy x, then x has a selective advantage over y. The strategy x is a local neighborhood invader strategy (Apaloo, 1997), if this condition holds for all y close to x. The interpretation of a neighborhood invader strategy is that such a strategy has a selective advantage against collective deviations to any (local) deviation strategy. As in the original definition the strategies x and y are one-dimensional parameters, y is ‘ϵ -close’ to x if |y − x| < ϵ . Eshel (1983) also considers collective deviations given some evolutionarily stable strategy x when asking

‘‘If a large enough majority of the population prefers a strategy y which is sufficiently close to x (. . . ), will it be advantageous for each individual in this population to choose a strategy closer to, rather than further apart from x?’’ ‘y sufficiently close to x’ is to be understood as |y − x| < ϵ . If a pure strategy x is a continuously stable strategy, the answer to this question is affirmative. Eshel (1983) also offers a necessary and a sufficient condition for continuous stability which involve the second derivative of the payoff function with respect to the strategy of the opponent. In the game studied here, the strategy of the opponent is the population strategy which is a function mapping valuations to bids and the necessary and sufficient conditions cannot be applied. Perhaps the strongest notion of dynamic stability in noncooperative games is evolutionary robustness (Oechssler and Riedel, 2002) as it requires robustness against small collective deviations and large deviations by single agents. Originally the concept was stated for mixed strategies. For the purpose of this paper it suffices to consider pure strategies. More precisely, a pure strategy x is evolutionarily robust if π (x, y) > π (y, y) for all pure strategies y different from x but close in the sense of both types of deviations. Clearly any evolutionarily robust strategy x is also a local neighborhood invader strategy. 3.2. Discrete versus continuous strategy sets Consider a game with two pure strategies ‘‘0’’ and ‘‘1’’. Suppose the current state is that all agents play ‘‘0’’. Let us briefly view two distinct deviations: Deviation A: a small fraction ϵ of agents deviate to the strategy ‘‘1’’. Deviation B: all agents deviate to the mixed strategy (1 − ϵ) · ‘‘0’’ + ϵ · ‘‘1’’. Let σϵi be the pure strategy played by a randomly chosen agent after the deviation i ∈ {A, B}. Then Prob(σϵA = ‘‘0’’) = 1 − ϵ = Prob(σϵB = ‘‘0’’) and Prob(σϵA = ‘‘1’’) = ϵ = Prob(σϵB = ‘‘1’’). The type of deviation is irrelevant for the payoffs of an individual. Consider now the continuous strategy set S = [0, 1] with the current state δ0 .2 Deviation A would correspond to the distribution σϵA = (1 − ϵ) · δ0 + ϵ · δ1 , where a small fraction of agents deviates to strategy 1 and deviation B would be σϵB = δϵ , where the whole population deviates to strategy ϵ close to strategy zero. How close are σϵA and σϵB to δ0 ? The answer depends on the measure of distance which is used. For two continuous functions f , g : R+ → R define

ϵxA = min{ϵ ≥ 0 : f (x) ≤ g (x) + ϵ and f (x) + ϵ ≥ g (x)} and

  x+ϵ ϵxB = min ϵ ≥ 0 : f (x) ≤ g (y)dy and x−ϵ   x+ϵ f (y)dy ≥ g (x) . x−ϵ

Define di (f , g ) = max{ϵxi : x ∈ R+ } for i = A, B. Then dA (δ0 , σϵA ) = ϵ

and dA (δ0 , σϵB ) = 1

and dB (δ0 , σϵA ) = 1

and

dB (δ0 , σϵB ) = ϵ.

Depending on the choice of the measure of distance, one kind of deviation is close to the equilibrium strategy while the other

2 δ is the Dirac measure on strategy x, the pure state in which all agents play x. x

L.P. Metzger / Journal of Mathematical Economics 50 (2014) 160–166

163

is not. If the standard definition of evolutionary stability is used for continuous strategies, then an evolutionarily stable strategy is stable against deviations that are close in the sense of dA . The concepts of continuously stable strategy and (local) neighborhood invader strategy use the word close in the sense of dB . A strategy is evolutionarily robust if it is stable against deviations that are close in the sense of either dA or dB , hence min {dA , dB }. The metric for evolutionary robustness uses





x+ϵ

g (y)dy + ϵ and ϵx = min ϵ ≥ 0 : f (x) ≤ x−ϵ  x+ϵ f (y)dy + ϵ ≥ g (x) ,

Fig. 2. Equilibrium strategies for distributions F and G.

x−ϵ

γ (v) to β(v) in payoff monotonic dynamics. I argue that for some valuations the equilibrium strategy β is neither a continuously

d(f , g ) = max{ϵx : x ∈ R+ } and is a simplified version of the Prohorov metric for the special case in which the preimage of the functions f and g is R+ . Note that dA (·) and dB (·) coincide on finite strategy sets. 3.3. A justification for B-deviations If F is uniform on [0, 1], the equilibrium strategy is β(v) = − ln(1 − v) − v for v ∈ [0, 1]. Imagine that f and F change to  1  1 − a + 4av if v < 2 and fa (v) =  1 + 3a − 4av if v ≥ 1 2  1 2  (1 − a)v + 2av if v < 2 Fa (v) =  (1 + 3a)v − a − 2av 2 if v ≥ 1

2 for a ∈ [0, 1]. If a = 1, then fa is the density of the sum of two variables that are uniform on [0, 21 ]. The equilibrium strategy changes to

 v 1 w(1 − a + 4aw)   dw if v <  2  1 − ( 1 − a )w − 2a w 2   0 1   2 w(1 − a + 4aw) βa (v) = dw  1 − (1 − a)w − 2aw 2 0     v  w (1 + 3a − 4aw) 1   dw if v ≥ .  + 1 1 − (1 + 3a)w + a + 2aw 2 2 2 If a = 1 then

β1 (v) =

    

v

1 − 2w 2

0

   

 0

4w 2

1 2

4w

dw

2

1 − 2w 2

dw +

v

 1 2

4w 1−w

dw

if v <

1

if v ≥

1

2 2

.

A sudden change from fa to f0 can then be seen as a B-deviation: given f0 the equilibrium strategy is β0 , but the population still plays the strategy βa . More generally, consider the situation illustrated in Fig. 2. Initially the valuations are distributed according to a distribution function G with density g and the population plays equilibrium v w g (w) strategy γ with γ (v) = 0 1−G(w) dw for all v ∈ V . If a small group of agents with fixed valuations enters the population, then the distribution of valuations changes slightly from G to F (and g to f ). This invasion by a small group is as if the whole population simultaneously and identically deviates to strategy γ which is and close to the equilibrium strategy β with β(v) = distinct v wf (w) dw for all v ∈ V . 0 1−F (w) In the sections below I investigate whether the conditions are met such that agents of any type v change their strategy from

stable strategy (Eshel and Motro, 1981), nor a (local) neighborhood invader strategy (Apaloo, 1997, 2006) and hereby that the conditions for evolutionary robustness (Oechssler and Riedel, 2002) and neighborhood superiority (Cressman, 2005) are not met. Maynard Smith (1974) shows for the war of attrition in which all agents have the same valuation  ∞v¯ that the mixed equilibrium strategy σ (b) = v1¯ e−b/¯v satisfies 0 σ (b)π (b, δm )db > π (m, δm ) for all pure strategies m ∈ R+ . Hence π (σ , δm ) > π (m, δm ) for all m ∈ R+ . Therefore we may conclude for the war of attrition with complete information that the mixed equilibrium is evolutionarily robust. Corollary 1. The fully mixed equilibrium in the war of attrition with complete information is evolutionarily robust. Remarkably, the war of attrition with complete information is the unique game known to the author that has an evolutionarily robust equilibrium in mixed strategies.3 Oechssler and Riedel (2001) show that the mixed equilibrium in the war of attrition with complete information is strongly uninvadable, whereas Bomze (1990) shows that any strongly uninvadable state is asymptotically stable in the replicator dynamics. I provide a brief discussion on the stability of the equilibrium in the war of attrition with complete and incomplete information at the end of Section 4.3 after having presented the formal results. 4. Propositions While the equilibrium in the war of attrition with complete information is mixed, the war of attrition with incomplete information has an equilibrium in pure strategies. Riley (1980) shows that this equilibrium is an equilibrium with unique best replies, if the number of agents in the population is sufficiently large. This observation offers an important insight: as the first condition of evolutionary stability, namely π (b, β) > π (b˜ , β), ∀b˜ ̸= b, is satisfied for all b, the second condition does not need to be checked. In what follows I explore whether there exists a strategy that violates local superiority, the second condition of evolutionary stability. 4.1. Synopsis Firstly I construct a strategy γ that is arbitrarily close to the equilibrium strategy β in the sense of that |β(v) − γ (v)| ≤ ϵ and intersects the equilibrium strategy once from below. Using this strategy Theorem 1 shows that the equilibrium strategy fails the criterion of continuous stability (Eshel and Motro, 1981). As local

3 I am grateful for the associate editor’s remark that the game with the payoff function f (x, y) = −x2 + a · x · y has an evolutionarily robust Nash equilibrium in pure strategies at (0, 0) for a < 1.

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L.P. Metzger / Journal of Mathematical Economics 50 (2014) 160–166

neighborhood stability (Apaloo, 1997) implies continuous stability in games with twice differentiable payoff functions defined on one-dimensional strategy spaces it is not surprising that the equilibrium strategy also fails to be a neighborhood invader strategy in this model, as Theorem 2 shows. The proof is necessary in the game studied here because the strategy of the opponent is a function rather than a one-dimensional real number. As any evolutionarily robust strategy (Oechssler and Riedel, 2002) is a neighborhood invader strategy, Theorem 2 implies that the equilibrium strategy is not evolutionarily robust. Lastly, I argue that the equilibrium strategy does not satisfy neighborhood superiority (Cressman, 2005) and hereby that there is no argument in favor of dynamic stability of the equilibrium in the replicator dynamics. Fig. 3. Equilibrium strategy β and deviation strategy γ .

4.2. A deviation strategy

v

w·f (w)

Given the equilibrium strategy β with β(v) = 0 1−F (w) dw , some small but positive ϵ and any valuation v¯ > 0 with F (¯v ) < 1 define valuation v˜ (ϵ) such that

β(˜v (ϵ)) = ϵ ·

v¯ − v˜ (ϵ) . v¯ + 1 − v˜ (ϵ)

we have lim

γ (v + h) − γ (v)

h↑0

h

= lim

(1)

ϵ

= β ′ (v) +

h↓0

(¯v + 1 − v)2 γ (v + h) − γ (v) h

∀ v ∈ (˜v , v¯ ).

Hence γ ′ (v) exists and γ ′ (v) > β ′ (v) for all v ∈ (˜v , v¯ ]. Lemma 1. v˜ (ϵ) < v¯ exists and is unique. Proof. As β(v) and v¯ −v  v¯ +1−v v=0



=

v¯ v¯ +1

v¯ −v v¯ +1−v

are continuous functions, β(0) = 0,

> 0, β(¯v ) > 0,

v¯ −v  v¯ +1−v v=¯v



v¯ −v ∂ v¯ + 1−v ∂v

v·f (v) 1−F (v)

= 0, ∂β(v) = ∂v

> 0 for all v ∈ (0, v¯ ] and = − < 0 for all v < v¯ + 1 the valuation v˜ (ϵ) exists, is unique and is strictly smaller than v¯ .  Lemma 2.

∂ v˜ (ϵ) ∂ϵ

1 (¯v +1−v)2

> 0 and limϵ→0 v˜ (ϵ) = 0.

Proof. As v˜ (ϵ) <

v (ϵ) v¯ is implicitly defined by β(˜v (ϵ))−ϵ · v¯ +v¯ −˜ 1−˜v (ϵ) 

=

 v¯ −v ∂ β(v)−ϵ· v¯ + 1−v ∂v

0 and > 0, the derivative of v˜ (ϵ) with respect to ϵ exists and is given by

   ∂ v˜ (ϵ) v¯ − v 1   = ·   ∂ϵ v¯ + 1 − v ∂ β(v)−ϵ· v¯ +v¯ −v 1−v  ∂v

v=˜v (ϵ) v¯ −˜v (0)

Hereafter the dependency of v˜ on ϵ is suppressed for notational convenience. Define the deviation strategy γ such that if v ≤ v˜ if v˜ < v ≤ v¯ + if v > v¯ +

1 2

As γ is differentiable and strictly monotonic for all v ∈ (˜v , v¯ ] the inverse γ −1 (b) exists and is differentiable and strictly monotonic   for all b > 0 such that γ −1 (b) ∈ (˜v , v¯ ]. Using γ γ −1 (b) = b and ∂γ (v) ∂v

> 0 we derive

∂γ −1 (b) ∂b

=

1 . ∂γ (v)  ∂v v=γ −1 (b)

4.3. Main results Suppose now that all agents choose their bids according to the population strategy γ whereas the equilibrium strategy is β . Theorem 1 states that for a positive share of individuals in the population it is advantageous to choose a bid that is further apart from rather than closer to the bid which is specified by the equilibrium strategy β . Theorem 1. For any ϵ > 0 there exists a population strategy γ : V → B with maxv∈V |β(v) − γ (v)| ≤ ϵ, γ ̸= β and a set of valuations V ⊂ V with F (V ) > 0 such that for some δ > 0

> 0.

Suppose that limϵ→0 v˜ (ϵ) > 0. Then β(˜v (0)) = 0 · v¯ +1−˜v (0) contradicts β(v) = 0 ⇔ v = 0. 

0    v¯ − v  γ (v) = β(v) − ϵ · v¯ + 1 − v    β(v) + ϵ



1 2

(2)

.

Fig. 3 illustrates the equilibrium and deviation strategies β and γ . Theorems 1 and 2 make use of the properties that γ is continuous and differentiable and that γ ′ (v) > β ′ (v) for all valuations v ∈ (˜v , v¯ ) and that |β(v) − γ (v)| ≤ ϵ for all v ∈ V . Lemma 3. γ (v) is differentiable and γ ′ (v) > β ′ (v) for all v ∈ (˜v , v¯ ]. Proof. As for all v ∈ (˜v , v¯ ] the deviation strategy γ (v) is defined v¯ −v by γ (v) = β(v) − ϵ · v¯ + and as β ′ (v) < ∞ ∀ v ∈ (˜v , v¯ ] 1−v

π (b|v, γ ) > π (γ (v)|v, γ ) ⇔ |β(v) − γ (v)| < |β(v) − b| for all bids b ∈ B with |γ (v) − b| < δ and all valuations v ∈ V . While the proof makes explicit use of the deviation strategy as defined in Eq. (2), any differentiable and strictly monotonic deviation strategy with steeper slope than the equilibrium strategy for an open interval of valuations around an intersection would serve. Proof. For any small ϵ > 0 consider v˜ as implicitly defined in Eq. (1) and the deviation strategy γ as defined in Eq. (2). The set V = (˜v , v¯ ) has positive measure since by Lemma 1 v˜ exists and v˜ < v¯ . Clearly β(v) − γ (v) = ϵ · v¯ +v¯ −v > 0 ∀ v ∈ V and for 1−v all v ∈ V β(v) − γ (v) ≤ ϵ . Hence β(v) > γ (v) ∀ v ∈ V and maxv∈V |β(v) − γ (v)| ≤ ϵ . An agent, who has valuation v , who bids b > 0 and who faces a population of agents who choose their bids according to the population strategy γ , expects payoffs

 γ −1 (b)  −1  π (b|v, γ ) = v · F γ (b) − γ (w) · f (w)dw   −1 v˜  − b · 1 − F γ (b) .

L.P. Metzger / Journal of Mathematical Economics 50 (2014) 160–166

By Lemma 3 we know that the slope of γ is steeper than the slope of β for all v ∈ V and hence

 ∂π(b|v, γ )  v · f (v) + F (v) − 1 =  ∂b γ ′ (v) b=γ (v) <

v · f (v) + F (v) − 1 = 0. β ′ (v)

By continuity of π (b|v, γ ) in b there is some δ > 0 such that for all b with |γ (v) − b| < δ

π(b|v, γ ) > π (γ (v)|v, γ ) ⇔ b < γ (v). As β(v) ≥ γ (v) ∀ v ∈ V we have b < γ (v) ⇔ |β(v) − γ (v)| < |β(v) − b| which establishes the claim.  If for any mass deviation to a population strategy γ close to β any agent were better off choosing a strategy closer to the equilibrium strategy β rather than further apart, there would be strong arguments in favor of the hypothesis that the equilibrium strategy is attracting. It is this property that defines a continuously stable strategy (Eshel and Motro, 1981). Theorem 1 states that the opposite is true for some arbitrarily close deviation strategy. There is an open set V of valuations such that further deviation from the equilibrium is advantageous. Note that the theorem holds for any ϵ > 0 and any valuation v¯ ∈ V . The mass of the set V increases if ϵ tends to zero and if v¯ increases in which case the deviation strategy gets closer to the equilibrium strategy. In particular, if v¯ approaches max V then by Lemma 2 we have that limϵ→0 F (V ) approaches 1! The destabilizing property identified by Theorem 1 does hold for a substantial part of the population. Corollary 2. The equilibrium strategy β of the war of attrition with private information is not continuously stable as defined in Eshel and Motro (1981). If the payoff function of a game is twice differentiable with respect to both players’ strategies, Oechssler and Riedel (2002) show that if a strategy is continuously stable, then it must be a local neighborhood invader strategy. In the game studied here, the strategy of the opponent is a function which makes it difficult to define the derivative. I offer an alternative proof to show that the equilibrium strategy β is no neighborhood invader strategy. Theorem 2. For any ϵ > 0 there exists a population strategy γ :

V → B with maxv∈V |β(v) − γ (v)| ≤ ϵ, γ ̸= β and a set of valuations W ⊂ V with F (W ) > 0 such that

165

By continuity of γ (·), β(·) and γ −1 (·), π (β(v)|v, γ ) and π (γ (v)| v, γ ) are continuous in v . Hence there exists a δ > 0 and an open set W ⊂ V with sup W = v¯ and v¯ − inf W < δ and F (W ) > 0 such that

π (β(v)|v, γ ) − π (γ (v)|v, γ ) < 0 ∀ v ∈ W , which establishes the claim.



Corollary 3. The equilibrium strategy β of the war of attrition with private information is neither a local neighborhood invader strategy as defined in Apaloo (1997) nor a neighborhood invader strategy as defined in Apaloo (2006). Corollary 1 states that the fully mixed equilibrium of the war of attrition with a common valuation is evolutionarily robust. From Milgrom and Weber (1985) we know that the distribution over bids of the pure strategy equilibrium in the game with private valuations converges to the distribution of the mixed strategy equilibrium in the game with a common valuation as the distribution of values converges to the degenerate distribution in which all agents have the same valuation. However, this strong connection between the two models does not carry over to the stability properties of the equilibria. As evolutionary robustness (Oechssler and Riedel, 2002) implies neighborhood invader strategy (Apaloo, 2006), the equilibrium strategy β cannot be evolutionarily robust. Corollary 4. The equilibrium strategy β of the war of attrition with private information is neither an evolutionarily robust strategy as defined in Oechssler and Riedel (2002) nor is it neighborhood superior as defined in Cressman (2005). If a tiny uncertainty is introduced into the model with one common valuation, the equilibrium strategy ceases to be evolutionarily robust. What is the explanation for this unintuitive phenomenon? While in the model with private valuations a collective deviation to some strategy γ was motivated by a change in the valuations of few agents, such a change is excluded by assumption in the model with one common valuation. Therefore, in a model with one common valuation the equilibrium cannot be destabilized by a change of the distribution of valuations. The introduction of uncertainty – even if arbitrarily small – changes the scenario abruptly. The instability result implied by Theorem 2 reveals the crucial importance of the assumption of whether or not valuations of the agents are subject to rare random perturbations.

π(β(v)|v, γ ) < π (γ (v)|v, γ ) ∀ v ∈ W .

4.4. Dynamic instability

Proof. For any ϵ > 0 consider the deviation strategy γ as defined in Eq. (2). maxv∈V |β(v) − γ (v)| ≤ ϵ, β(v) ≥ γ (v) ∀ v ≤ v¯ and γ (¯v ) = β(¯v ). The difference in payoffs of an agent with valuation v facing the population strategy γ who chooses β(v) rather than γ (v) is given by

Hofbauer et al. (1979) show that in games with finitely many strategies any evolutionarily stable strategy (Maynard Smith, 1974) is asymptotically stable in the replicator dynamics (Taylor and Jonker, 1978). They also give an example of an asymptotically stable state that is not represented by an evolutionarily stable strategy, showing that evolutionary stability is sufficient but not necessary for dynamic stability. As Bishop et al. (1978) show that the unique equilibrium of the war of attrition with private valuations is evolutionarily stable, one could conclude that it is dynamically stable—were it an equilibrium of a game with finitely many strategies. This is not the case, however, as for each valuation the set of potential bids is the set of non-negative reals. Oechssler and Riedel (2002) show that the stronger concept of evolutionary robustness offers a sufficient condition for dynamic stability in doubly symmetric games with infinite strategy spaces. Cressman (2005) shows that neighborhood superiority is a sufficient condition for dynamic stability in general games with infinite strategy spaces. With Theorems 1 and 2 I show that the equilibrium in the war of attrition with private valuates violates necessary conditions for evolutionary robustness and neighborhood superiority. This paper does not provide a rigid proof for dynamic instability.

π(β(v)|v, γ ) − π (γ (v)|v, γ )     = v · F γ −1 (β(v)) − F (v) −

γ −1 (β(v))

 v

γ (w) · f (w)dw

− β(v) · (1 − F (γ −1 (β(v)))) + γ (v) · (1 − F (v)). Since γ −1 (b) is strictly increasing in b and β(v) ≥ γ (v) for all v ≤ v¯ we have that γ −1 (β(v)) ≥ v for all v ≤ v¯ . As γ −1 (β(¯v )) = v¯ and as by Lemma 3 γ ′ (¯v ) > β ′ (¯v ) the derivative of the payoff difference with respect to v evaluated at v = v¯ is positive:

 ∂ (π(β(v)|v, γ ) − π (γ (v)|v, γ ))   ∂v v=¯v 1 − F (¯v ) ′ ′ 2 = · (γ (¯v ) − β (¯v )) > 0. γ ′ (¯v )

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L.P. Metzger / Journal of Mathematical Economics 50 (2014) 160–166

violates continuous stability (Eshel and Motro, 1981) and (local) neighborhood invader strategy (Apaloo, 1997, 2006) for an open set of valuations. The share of these valuations increases as the level of uncertainty diminishes. As a consequence the equilibrium strategy does not satisfy evolutionary robustness (Oechssler and Riedel, 2002) or neighborhood superiority (Cressman, 2005). I expect that the evolutionarily stable strategy is not attracting in the replicator dynamics. References

Fig. 4. V = [0, 1], F (v) = v, ϵ = 0.4, v¯ = 0.6 → v˜ = 0.349.

Fig. 5. V = [0, 1], F (v) = v, ϵ = 0.1, v¯ = 0.9 → v˜ = 0.254.

5. Numeric calculations The discussion of Theorem 1 revealed that the set V = (˜v , v¯ ) of valuations for which even further deviations from the equilibrium strategy are preferable becomes an arbitrarily large subset of V if ϵ tends to zero and v¯ tends to the upper bound of V . Theorem 2 identifies a small open subset W ⊂ V such that the difference in payoffs of the equilibrium strategy β against deviation strategy γ and γ against itself is negative. Theorem 2 does not offer any analytic insight into whether or not the set W increases or decreases in the same manner as the set V . Numeric calculations indicate that the sets W and V = (˜v , v¯ ) are equal, as Figs. 4 and 5 illustrate. 6. Conclusion In this paper I analyze the dynamic stability of equilibria in second price all-pay auctions with continuous bids and incomplete information on the valuation of the opponent. This is of particular interest as Bishop et al. (1978) show the existence of a unique evolutionarily stable strategy but Bishop and Cannings (1978) show convergence only for finite strategy sets. This paper aims at explaining the gap. In games with finite strategy sets an evolutionarily stable strategy can neither be invaded by independent and undirected deviations nor by close collective deviations. Bishop et al. (1978) test their evolutionarily stable strategy only against independent deviations. I give a plausible interpretation of mass deviations which is valid for all Bayesian games. According to this interpretation a mass deviation is triggered by a change of valuations of an arbitrarily small fraction of the population. I show that there exists a deviation strategy that

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