Asymmetric Contests: A Resolution of the Tullock Paradox* John G. Riley** Revised January 12, Forthcoming in

Transcription

1 Asymmetric Contests: A Resolution of the Tullock Paradox* by John G. Riley** Reised January, 998 Forthcoming in "Money, Markets and Method: Essays in Honour of Robert W. Clower" eds Peter Howitt, Elisabetta De Antoni and Axel Leijonhufud *This research was supported by a grant from the National Science Foundation. Its origins can be found in an unpublished paper by Hirshleifer and Riley [978]. The comments and suggestions of Peter Howitt, Son-Ku Kim and Michael Waldman are gratefully acknowledged. **Department of Economics, UCLA

2 Starting with the work of Tullock [980], there has been a continuing stream of articles deeloping the theory of "rent seeking" expenditures in the competition for a political prize (such as a franchise or tariff protection). Typically rent seeking contests hae been modeled as a one shot game in which each participant incurs some nonrefundable costs in the pursuit of the prize. Under the assumption of symmetric players, the central finding of this research is that entry will take place until equilibrium expenditures are, in expectation, equal to the alue of the prize. If all these expenditures are on socially unproductie actiities (e.g. paying lobbyists to compete with other lobbyists), the entire alue of the goernment opportunity is then dissipated. Unfortunately, as Tullock [980] himself noted, this appears to explain far too much. While the rent-seeking industry certainly thries at all leels of goernment, the expenditures incurred by rent-seekers seem small compared with the prizes inoled. The challenge then, is to explain why rent-seeking contests typically inole expenditures which fall far short of the alue of the prize. A first step towards this goal was undertaken by Hillman and Riley [989] who argue that in many cases there are considerable asymmetries between the payoffs to different contestants. Gien these asymmetries, the equilibrium expected expenditures tend to be determined largely by the second highest payoff. Thus the contest generates a positie expected surplus. In this paper it will be argued that much larger surpluses are obtained if the competition has elements of a war of attrition. Indeed the more closely that a contest resembles To the extent that rent seeking results in a pure transfer of resources to a public decisionmaker, this creates a greater incentie for indiiduals to compete for the position of decisionmaker. Dissipation thus occurs one step remoed from the actual decision-making. Indeed in court cases inoling illegal rent-seeking (bribes) is it remarkable how small

3 a "war of attrition" the greater is the gap between expected total expenditure and the high (or second) aluation of the prize. Moreoer, if a contest does hae strong elements of a war of attrition (where the winner and loser end up paying nearly the same) the ratio of expected total expenditure to the alue of the prize is low unless the asymmetry in aluations is ery small. Political rent-seeking is just one of many rialrous contests that hae been studied by economists. For example, with declining demand, there will typically be an incentie for firms to exit from an oligopoly (Ghemawat and Nalebuff [985], Fudenberg and Tirole [986], Whinston [988]). Similarly, in the race to inent, exit will eentually occur if the returns to gaining a patent decline with the length of the race. The results of this paper shed additional light on these contests as well.. The Model In this section we model a family of contests in which two agents compete. Both agents incur expenditures in the pursuit of the prize. The winner of the contest is the agent who spends the most. 3 In one extreme ersion of the model, all expenditures are sunk. Translating the description of the model into the language of auction theory, each bidder loses his entire "bid." At the other extreme, both agents must pay the lower of the two bids. This is the war of attrition. In this contest, the agents compete by spending at the same fixed rate and the winner is determined by the withdrawal of his opponent. The winner then incurs essentially the same cost as the loser. In the family of contests considered here, it will be assumed that the payment by the agent with the high bid is a conex combination of the two bids submitted. the apparent payments are compared with the potential pay-off. 3 In the case of a tie, the winner is determined randomly.

4 as follows Let b H be the high bid and b S be the second bid. The cost to each of the agents is then () Winner's payment = ( λ) bh + λb 0 λ S Loser's payment = b S. With λ = 0, this becomes the contest in which all bids are paid. With λ = it becomes the war of attrition. Let i, i =, be the alue of the prize to player i. If these aluations are the same, the game is symmetric. We can then appeal directly to the reenue equialence theorem which tells us that expected total expenditures are independent of the rules of the game. 4 In particular they are equal to expected total expenditures in an open auction in which the high bidder pays the second highest price. It is well known that the dominant strategy equilibrium of this game is for each agent to stay in until the asking price reaches his aluation. Thus expected total expenditures are equal to the alue of the prize. We now introduce asymmetry by assumi ng that agent 's aluation is higher, that is, >. This is common knowledge. As a preliminary we establish that, in equilibrium, no agent will submit a strictly positie bid amount with positie probability. Lemma : Continuity of the equilibrium mixed strategies With nonrefundable bids, as gien by (), no agent will bid b > 0 with positie probability. 4 While the reenue equialence theorem is usually presented with types continuously distributed, (see for example, Riley and Samuelson [98]) the results hold for finite type spaces as well. 3

5 Proof: Suppose agent bids $ b with strictly positie probability. If $ b <, agent 's probability of winning increases discontinuously at $ b while his expected cost increases continuously. Thus there must be some interal [ b $ δ, b $ ] such that agent is strictly better off bidding just aboe b $ than bidding in this interal. On the other hand, if b $ agent 's best response is to bid zero. Thus, in either eent, there is some interal [ b $ δ, b $ ] within which agent will not bid. But, gien that this is the case, agent 's bid of $ b is neer a best response since agent is strictly better off bidding $ b δ. That is, bidding $ b with strictly positie probability cannot be an equilibrium bidding strategy for agent. Arguing symmetrically, this must also be the case for agent. Q.E.D. Gien Lemma, we suppose, henceforth, that agent i adopts the mixed strategy ~ b i with c.d.f. Gi ( b), i =,. Applying arguments similar to those in Maskin and Riley [995], these equilibria are further characterized in Lemma. Lemma : Characterization of equilibrium bidding Agent i's bidding strategy, b ~ i, has support [ 0, z( λ )]. Moreoer at most one agent bids zero with positie probability. Proof: See appendix Before moing to a more formal analysis of the general case, we describe the equilibria in the two polar cases. 4

6 Proposition : Equilibrium in contests in which all bids are paid If all bids are paid, both agents adopt uniform mixed strategies on [0, ] conditional upon deciding to enter. Agent, with the high aluation always enters. Agent randomizes, entering with a probability equal to the ratio of the two aluations. Proof: Hillman and Riley [989] analyze this model in detail and establish that this is the unique equilibrium. Here we simply confirm that the strategies are best replies. Consider a bid of b [ 0, ] by agent. With agent bidding according to the continuous c.d.f G ( b ), agent 's expected payoff is: U = G ( b) b b b If G ( b) =, then U = ( ) b = 0. Thus agent is indifferent between staying out of the contest and bidding on the interal [0, ]. Next suppose agent enters the contest with probability p and, if he does, adopts the same uniform mixed strategy. Agent then faces a mixed strategy which can be represented by the c.d.f. G b p p b ( ) = +. His expected payoff is therefore: If p U = [ p + p b ] b =, then U = [ ] = Thus agent is also indifferent as to bids in the interal [0, ], as required for a mixed strategy equilibrium. Finally, we note that since agent wins with probability by bidding, he is strictly worse off making any higher bid. 5

7 Q.E.D. Note that the agents' equilibrium payoffs ( U E E, U ) = (, ) are exactly as in the 0 open auction. For in the latter, the price is bid up to if agents follow their dominant strategies. Howeer, in a nonrefundable contest, with both agents bidding according to mixed strategies, it is no longer the case that the high alue bidder wins almost surely. It follows that the gross benefit generated by the contest is strictly less than. But expected expenditures are just the difference between the gross benefit and the sum of the agents' expected payoffs. It follows that expected expenditures are less than. These expenditures are easily calculated. The expected expenditure for the uniform bidding strategy is just while agent adopts it with probability p total expenditure is therefore + ( )( ) In what follows it will proe helpful to define. In equilibrium, agent adopts this strategy with probability =, otherwise he does not compete. Expected µ = asymmetry ratio = and R( λ, µ ) expected total expenditure. Then, from the aboe analysis: + µ () R( 0, µ ) = ( ), where µ is the probability that agent enters. 6

8 Holding constant and letting grow large so that µ 0, agent 's probability of entering the contest approaches zero, while agent 's behaior is unchanged. Expected total expenditure then approaches all bids paid is half the second aluation.. Thus in the limiting case, total expected expenditure in the contest with Turning now to the war of attrition, suppose that units of time are measured so that each agent spends $ per unit of time he remains in the contest. A bid of b i by agent i is therefore a commitment to stay in the contest until time b i. As we shall see, one of the special features of the war of attrition is that there is a continuum of Nash (sub-game perfect) equilibria. 5 Proposition : The war of attrition For the war of attrition there is a continuum of (perfect) Nash mixed strategy equilibria. The equilibrium bid distributions hae c.d.f.'s G ( b), G ( b) satisfying (3) R S T b G ( b) = A e, A, A b G ( b) = A e, Max{ A, A } = Proof: Suppose that the war of attrition has lasted until time b. The conditional probability that agent will withdraw in the next interal of length b is G ( b + b) G ( b) G ( b) 5 In this paper agents hae full information about their opponents aluations. But with priate information as well, there is a continuum of equilibria in the war of attrition. (Riley, [98], Nalebuff and Riley, [985]) 7

9 If agent stays in the contest the additional time his cost increases by b. His expected net payoff to doing so is therefore U G ( b + b) G ( b) = { } G ( b) b Diiding by b and taking the limit as b 0 we obtain: du db ' G ( b) = { G b ( ) } In a mixed strategy equilibrium, agent must be indifferent between bidding b and b + b. Therefore du = 0. Since exactly the same argument holds for agent we hae the following db two equilibrium conditions. ' G ( b) = and G ( b) G ' ( b) = G ( b) Integrating these two equations yields (3). Note that Gi ( b) is the probability that agent i spends a positie amount in the contest. Thus, from (3), A i must lie in the interal [0,]. We now argue that Max{ A, A } =. Suppose not. Then, from (3), G ( 0) and G ( 0) are both strictly positie. But then equilibrium payoffs of both agents are strictly positie, since there is a strictly positie probability of winning with any ery small bid ε. It follows that neither agent will eer decline to enter the contest so G i ( 0 ) must be interpreted as a strictly positie probability of bidding zero. But this contradicts Lemma. Q.E.D. We now turn to an examination of the expected expenditures in this contest. From (3), the higher is A i, the lower is Gi ( b) and hence the more aggressiely agent i bids. Then 8

10 expected total expenditure must be increasing in A i. At the lower limit with Min{ A, A } =, one agent neer bids and so the other wins at a cost of zero. Total 0 expenditure is therefore zero. At the upper limit with A = A =, Gi ( 0) = 0, i =,. Then both agents hae an equilibrium expected payoff of zero. It follows that expected total expenditure must be equal to the expected gross benefit generated by the contest. Let p be the probability that agent wins. Then expected total expenditure is p + ( p ). If agent bids b he wins with probability G ( b). Taking the expectation with respect to b, z agent 's probability of winning is p = G ( b ) dg ( b ) 0 Making use of (3) it is fairly straightforward to confirm that expected total expenditure is equal to +. Since = µ, this can be rewritten as. Noting again that this is the + µ special case with λ =, expected total expenditure must satisfy R(, ) [ 0, µ ]. This + µ Expected total expenditures R(, µ ) = + µ O µ = Figure : Range of expected total expenditures in the war of attrition 9

11 range is depicted in Figure. For each µ =, eery point below the cure R( 0, µ ) = + µ represents expected total expenditure from some equilibrium. Except in the symmetric case, with µ =, equilibrium expected total expenditures can exceed the second aluation.. A Family of Nonrefundable Contests In the first of the two polar cases examined aboe, the full bid is lost. In the second, there is no fixed cost. The actual cost incurred by the winner ranges from zero to his own bid. We now consider the equilibrium for contests in which the actual cost incurred has a fixed and ariable component. From (), if agent bids b and his opponent bids b less than b, agent 's net payoff is ( λ) b λ b. If his opponent bids more than b, agent pays his bid. From Lemma, there can be no mass points for b > 0 therefore the probability of a tie is zero. With agent 's bidding strategy ~ b distributed according to the c.d.f G ( ), it follows that agent 's expected payoff is zb (4) U( b) = [ ( λ) b λb ] dg ( b ) ( G ( b)) b 0 But, for agent 's equilibrium strategy to be mixed, with support [0, z], U ( b) must be constant oer this interal. Differentiating (4) by b we therefore obtain: ' ' U ( b) = G ( b) ( λ G ( b)) = 0, b [ 0, z]. Since exactly the same argument applies for agent, the equilibrium c.d.f.'s must satisfy: (5) R S T ' G ( b) = λg ( b) ' G ( b) = λg ( b). These two differential equations can be rewritten as follows. 0

12 ' λg i λ =, i =,, j i. λg ( b) i j Integrating we therefore obtain: λb j (6) λg ( b) = Ae, j i, i =, i i Since 0 G ( b ) i, it follows that A i [ 0, ]. Moreoer, from Lemma, at most one agent bids zero with positie probability. Therefore Min{ G ( 0), G ( 0)} = 0 and so Max{ A, A } =. At the upper support z, Gi ( z) =, i =,. Substituting this into (6) we obtain: λz λz (7) A e = λ = A e = λz A [ e ] Therefore, λ ( λ) = A[ ] A If is strictly greater than it follows from (7) that A it follows from (7) that A established that R S T > A. If = = A. Since the larger of A and A is, we hae therefore (8) A A =, = ( λ ), <., = Together (6), (7) and (8) define the unique Nash equilibrium. To summarize, we hae proed:

13 Proposition 3: Equilibrium in the general contest Suppose 0 λ <. Then there exists a unique Nash equilibrium. The equilibrium bidding strategies satisfy R S T λb A e λb λg ( b) = λg ( b) = e, b [ 0, z] where z satisfies (7) and A is gien by (8). Before turning to an examination of expected expenditures, it is of interest to contrast Propositions and 3. In the former, there is a continuum of equilibria, while in the latter equilibrium is unique. Since the pure war of attrition is the limiting case polar case we can use Proposition 3 to ask which of the continuum of equilibria is selected in the limit as λ. Case : Identical aluations From (8), if aluations are identical so that µ = =, then A = A = and the unique equilibrium is the symmetric equilibrium. But in a symmetric equilibrium G ( 0) = G ( 0) and, from Lemma we know that at least one agent bids more than zero with probability. Therefore G ( 0) and G ( 0) must both be zero. It follows that the expected payoff to each buyer is zero and so expected total expenditure, R( λ, ) =. From Proposition, there is a continuum of equilibria when λ =, with expected expenditure lying anywhere between 0 and. What we hae established therefore, is that in the symmetric case taking the limit as λ approaches results in the selection of the equilibrium which has the greatest expected expenditures. 6 6 Of course this also implies that it is the most inefficient equilibrium which is selected.

14 As Nalebuff and Riley [985] hae shown in their analysis of the war of attrition with priate information, a similar result obtains with a different perturbation of the model 7. Specifically, they assume that each agent beliees that his opponent plays a Nash strategy with probability p <. With probability (- p) his opponent plays the "stubborn" strategy of competing foreer. For all p < there is a unique equilibrium which is symmetric and, as p approaches expected total expenditures approach the maximum. Case : Asymmetric contests A ery different result holds for the case of asymmetric aluations. If µ <, then from (8), as λ approaches the parameter A approaches zero rather than. From Proposition 3, it follows that the c.d.f. for the bid distribution of agent, G ( b) approaches for all b. That is, the probability that agent makes a positie bid approaches zero. It follows that agent wins almost immediately with probability approaching. Therefore, in the limit as λ, expected total expenditures decline to zero. Summarizing, we hae proed: Proposition 4: Equilibrium selection for the war-of-attrition If aluations are identical ( = ), expected total expenditures in the general nonrefundable contest approach as λ approaches. If aluations differ ( > ) expected total expenditures approach zero as λ approaches. Proposition 4 suggests a quite remarkable distinction between the case of symmetric and asymmetric aluations. With only the smallest of differences in aluations, the limiting equilibrium is for the agent with the lower aluation to concede immediately. The agent with the higher aluation then wins at a cost which approaches zero in the limit. Viewing such 7 An earlier closely related result can be found in Wilson [983]. 3

15 contests as social contests in which the expenditures are lost to society rather than transfers to some third party (as is the case in auctions), the Proposition suggests that war-of-attrition like contests are highly efficient mechanisms for resource allocation, except in the special case of complete symmetry. Howeer, we now argue that such a conclusion, while mathematically correct, is ery misleading. If agent bids b his probability of winning is G ( b). Taking the expectation oer b, agent 's expected probability of winning is z p G ( b) dg ( b) G ( 0) [ G ( b) G ( 0)] dg ( b) z z = = + 0 Similarly, agent 's probability of winning is z 0 p =z z G ( b ) dg ( b ) 0 The expected gross benefit from the contest is the expected aluation of the winner, that is, p + p. Agent i's equilibrium payoff is just the probability of winning with a zero bid times his aluation, that is G ( 0), j i. From Proposition 3, G ( 0) = 0 so buyer 's expected j payoff is zero, while agent 's expected payoff is G i ( 0). Then subtracting expected buyers' payoffs from the expected gross benefit, expected total expenditure is: R( λ, µ ) = p + p G ( 0) z( zλ ) z( zλ ) = [ G( x) G( 0)] dg( x) + G( x) dg ( x) µ 0 Note that z( λ ) is defined implicitly by (7) and µ is the asymmetry ratio. We next consider expenditure as a function of both λ and µ, oer the unit square depicted in Figure. 0 4

16 Ratio of aluations µ equal aluations All bids paid War of attrition O extreme inequality of aluations λ Contest parameter Figure : Family of Contests We begin by considering R( λ, µ ) near the boundaries. (a) 0 λ <, µ From (8), lim A. Therefore lim G ( 0 ) = 0. Also G 0 0 ( ) =. But the equilibrium = µ µ E expected payoff of buyer i is U = G ( 0), j i, i =,. Therefore, as the difference in the i i j aluations grows small, the sum of the equilibrium expected payoffs to the buyers approaches zero. As already noted, expected total expenditures are the difference between the expected gross benefits generated by the contest and the sum of the expected payoffs to the buyers. But, in the limit as the difference in the aluations approaches zero, the gross benefit approaches. Then expected total expenditures must approach, that is lim R( λ, µ ) = µ 5

17 It is readily confirmed that this result holds at the limit as well. That is, for all λ, R( λ, ) =. This is just another example of the general "equialence theorem" for symmetric contests. (b) λ = 0 From Proposition we know that with λ = 0, expected total expenditure increases linearly with µ. (c) λ From Proposition 4, for all µ <, expected total expenditure approaches zero as λ approaches. Now consider points in the interior of the unit square. From Proposition 3, λ λg = ( λ)( λ) [ e ] λg = ( λ)[ ] λ Also, from Proposition 3, G b ( ) is independent of and hence of µ. Therefore, the right hand side of the aboe expression is increasing in and hence decreasing in µ. Thus G b ( ) decreases as µ increases and so agent becomes more aggressie as µ increases while the behaior of agent is unchanged. It follows that expected total expenditure falls as the ratio µ = falls. It is not a simple matter to determine analytically how expected total expenditure aries with λ. Howeer, it is an easy matter to compute alues of R( λ, µ ) for arious alues of λ and µ. Tables and summarize a number of these computations. 6

18 Note that the top row, with λ = 0, is the limiting case as the contest rules approach the pure war of attrition (the nondiscriminatory contest). As we hae seen, expected total expenditure is zero in this limiting case. From the tables it is clear that, for any gien asymmetry ratio µ, expected total expenditure declines with λ. Thus the more a contest is like a war of attrition, the lower is the ratio of expected total expenditure to contestants' aluations. There are two ery striking features of these tables. If aluations are similar, expected total expenditure is high relatie to aluations, een when the contest is extremely close to the pure war of attrition. For example, with µ = = 0.9, expected total expenditure drops from 85% of the high aluation to 56% as λ increases from 0. to Thus when aluations are TABLE : Expected Total Expenditure as a Percentage of the High Valuation λ µ

19 TABLE : Expected Total Expenditure as a Percentage of the Second Valuation λ µ similar, competition bids away much of the alue of the prize, regardless of the rules of the contest. On the other hand, if aluations differ considerably, a ery differ ent conclusion emerges. For example if µ = = 0.5 expected total expenditure declines from 36% of the high aluation to only 4% as λ increases from 0. to 0.9. And if the asymmetry ratio is less than 0., expected total expenditure becomes a small fraction een of the second aluation. We summarize these conclusions in the following Proposition. 8

20 Proposition 5: Expected Expenditure in a Modified War of Attrition Suppose that the war of attrition is modified to allow for a part of the cost of competing to be proportional to each agent's own bid. If aluations are similar, expected total expenditure remains a ery significant fraction of the second aluation, een when these additional costs are extremely close to zero. On the other hand, when aluations are ery different, expected total expenditures are a small fraction of the second aluation. 4. The Tullock Paradox Resoled We will now argue that the results of the preious section can be used to explain why expenditures in rent seeking contests fall far short of the alue of the prize. While some of the expenditures made pursuing a political prize are "fixed" in nature, that is, independent of the bids of opponents, it is also the case that political competition is played out oer time and that it is possible to monitor the actiities of major opponents. Moreoer, once an opponent's budget is exhausted, the probability of ictory surely rises rapidly. To the extent that this is the case, a "rent seeking" contest takes on some of the characteristics of a war of attrition, rather than a pure "all bids lost" contest. 8 Consider then, a simple model of rent seeking in which each of two agents compete by spending resources at the same rate θ. In addition to the resources used up in the actual competition, if an agent prepares to stay in the contest until time t, he incurs a "fixed cost" of 8 Indeed, the more that politicians' war chests are funded by lobbyists, the greater is the incentie for a legislature to delay a decision, in the expectation that more funds will be forthcoming. 9

21 φ per unit of time until t. The competition continues until one agent withdraws. The other then becomes the winner. Consider the payoff to agent if he plans to stay in the contest until time b and his opponent concedes at time x < b. If the interest rate is r, the cost to agent is z z e θdt + e φdt = θc x + φc b x b rt = rt 0 0 rt where C( τ) z e dt. τ 0 ( ) ( ) C( τ ) is just the present alue of spending per period until time t. Let be the alue of the prize (discounted to the time of its receipt) and let the strategy of agent be represented by the c.d.f. G ( b). The present alue of staying in the contest until time b is then b z 0 rx U = [ e θc ( x) φc ( b)] dg ( x) ( θ + φ) C( b)( G ( b)) As in the preious section, we differentiate by b and then note that U ( b) must be constant in a mixed strategy equilibrium. We hae: ' rb ' U ( b) = e [ G ( b) θg ( b) ( θ + φ) = 0 Rearranging, this can be rewritten as ' θ G( b) = G ( b) θ + φ θ + φ Arguing exactly the same way for agent we also hae θ G ' ( b) = G ( b) θ + φ θ + φ 0

22 Comparing these two conditions with (5) it follows immediately that if we choose λ = θ θ + φ all the results of the preious section hold. In particular, if aluations differ mildly and θ (and hence λ) is sufficiently large, the equilibrium expected present alue of rent seeking expenditures will be a significant fraction of the second aluation but fall well short of this aluation. On the other hand, when aluations differ considerably, the agent with the high aluation wins at a low expected cost. 5. Concluding Remarks We hae demonstrated that the outcome of bidding in nonrefundable contests is highly sensitie to simplifying assumptions. We hae further argued that more plausible implications emerge if the assumptions of perfect symmetry and of no "fixed costs" are relaxed. Perhaps most importantly, we establish that when aluations differ and there are een ery small fixed costs in a modified "war of attrition", there is likely to be a ery significant degree of competition, although considerably less than under perfect symmetry. This conclusion contrasts dramatically with the implications of the model when there are no fixed costs. For, in this polar case, the contest is, with probability, oer as soon as it starts, and the high aluation agent is the winner. The theoretical results are then applied to a simple model of political "rent seeking". We argue that our model explains both the empirical eidence of a igorous and expensie battle for political faors and, at the same time a leel of total spending which falls far short of the alue of the political prize.

23 As it stands, our model, whether applied to political or any other form of competition, is highly stylized. While not demonstrated here, we conjecture that the equilibrium for the two agent case is also the unique equilibrium when there are more than two agents and the others hae strictly lower aluations. 9 Thus our model cannot explain nonrefundable contests with entry by three or more agents. There are two natural modifications of the model which will accommodate entry by more than agents. First, rather than assume that information about an opponent's aluation is perfect, it is more reasonable to assume that this information is, as least to a degree, priate. Agent three may know that agents and are more likely to hae higher aluations and agents and may know this also. Despite this, agent 3 will hae an incentie to enter if his priate aluation is sufficiently great, relatie to expectations. A second natural modification of the model is to allow for uncertainty about the effectieness of each agent's spending campaign. Then, een if an agent knows that her aluation is lower than two or more opponents, she may not be dissuaded from entry because of the chance that her campaign will hae a higher payoff per dollar inested. These issues will be explored in later work. Of course the richer model will yield a wider range of possible outcomes. Howeer, it is conjectured that, unless the model of nonrefundable contests is drastically modified, the basic conclusions characterized here will continue to hold. With these generalizations we will also be able to address directly the puzzling difference between the results of Fudenberg and Tirole [986] and Ghemawat and Nalebuff [985]. The former consider an exit game as a war of attrition. Each firm would rather continue as a 9 Indeed it is easy to confirm that if the two agents with the highest aluations bid as in Proposition 3, the best reply of any other agent with a lower aluation is to remain out. That is,

24 monopoly rather than quit the market, but both lose money as duopolists competing for a shrinking customer base. Fudenberg and Tirole show that with priate information and symmetry, the expected cost of the contest is equal to the expectation of the second highest present alue of remaining is the monopolist. (This is just an application of the Reenue Equialence Theorem.) Ghemawat and Nalebuff consider the exit decision under the assumption that one firm knows that the other has greater financial reseres to dip into. They conclude that the firm which does not hae the deep pocket will exit immediately. Generalizing the model analyzed her to allow for priate information will make it possible to better understand the releance of these strikingly different conclusions. Finally, starting with Maynard Smith [974] sociobiologists hae used the symmetric war of attrition to explain animal behaior. Gien the striking contrast between the symmetric and asymmetric cases, a richer model may be helpful in explaining less extreme outcomes than suggested by the current theory. Proposition 3 characterizes an equilibrium of the multi-agent game. 3

25 References Fudenberg, Drew and Jean Tirole (986), 'A Theory of Exit in Duopoly', Econometrica, (54), Ghemawat, P. and Barry Nalebuff, (985), 'Exit', RAND Journal of Economics, (6), Hillman, Arye and John G. Riley (989), 'Politically Contestable Rents and Transfers', Economics and Politics, (), Hirshleifer J. and John G. Riley (978), 'Elements of the Theory of Auctions and Contests', UCLA Department of Economics Working Paper 8B. Kennan, John and Robert Wilson (993), 'Bargaining with Priate Information', Journal of Economic Literature, (3), Maskin, Eric and John G. Riley (995a), 'Asymmetric Auctions'. UCLA Working Paper. (995b), 'Uniqueness of Equilibrium in High Bid Auctions' UCLA Working Paper Maynard Smith, John (974), 'The Theory of Games and the Eolution of Animal Conflicts', Journal of Theoretical Biology, (47), Nalebuff, Barry and John Riley (985), 'Asymmetric Equilibria in the War of Attrition'. Journal of Theoretical Biology, (3), Riley, John (980), 'Strong Eolutionary Equilibrium and the War of Attrition' Journal of Theoretical Biology, (8), Riley, John and William F. Samuelson (98), 'Optimal Auctions', American Economic Reiew, (7), Tullock, Gordon (980) "Efficient Rent Seeking" in ( eds), J.M. Buchanan, R. Tollison and G. Tullock, Toward a Theory of the Rent Seeking Society, Texas A&M Press,

26 Vickrey, William (96), ' Counterspeculation, Auctions and Competitie Sealed Tenders', Journal of Finance, (6), Whinston, Michael D. (988), 'Exit with Multiplant Firms', RAND Journal of Economics, (9), Wilson, Charles A. (983), 'Games of Timing with Incomplete Information', unpublished. 5

27 APPENDIX Lemma : Characterization of Equilibrium bidding in the general nonrefundable contest. Agent i's bidding strategy, b ~ i, has support [ 0, z( λ )]. Moreoer at most one agent bids zero with positie probability. Proof: Let z be the upper support of agent 's bid distribution. With no mass points (except possibly at b=0), agent wins with probability by bidding z. Then agent has no incentie to eer bid more than z and so z z z. But an identical argument holds for agent so = z. Let this common upper support be z( λ ). We hae already established that Gi ( b) is continuous on (0,z]. Suppose that there is some subinteral [ α, β ] of (0,z] oer which agent bids with zero probability. Let b $ be the smallest bid aboe b made by agent. Then agent 's probability of winning is constant on [ α, b $ ] and so agent is better off bidding α than anywhere in ( α, b $ ]. It follows that agent 's probability of winning is constant on ( α, b $ ]. But then bidding b $ is not, after all, an equilibrium best response for agent. It follows that there can be no interal [ α, β ] oer which agent does not bid. Thus the support of agent 's bid distribution is [ 0, z( λ )]. Of course the same argument applies to agent. Finally, if G ( 0) and G ( 0) are both positie, both agents win with positie probability by making an arbitrarily small bid. Then, equilibrium expected payoffs of both agents are positie and both will enter the contest with probability. It follows that Gi ( 0), i =, is the probability of a bid of zero (rather than the probability of staying out of the contest.) But if 6

28 agent bids zero with positie probability, agent is better off aoiding the risk of a tie and bidding some ε > 0 rather than bidding 0. Q.E.D. NOTE FOR TYPESETTER Greek letters used λ, µ, τ, θ, φ Figure should go at or close to three lines before the end of section. Figure should go on or near line 3 on page 5. 7