all-paycontestsfaculty.haas.berkeley.edu/rjmorgan/phdba279b/phdba297b_4...1 introduction...

All-Pay Contests(Ron Siegel; Econometrica, 2009)

PhDBA 279B13 Feb 2014

Hyo (Hyoseok) KangFirst-year BPP

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Outline

1 IntroductionAll-Pay ContestsAn Example

2 Main AnalysisThe ModelGeneric ContestsNon-generic ContestsParticipation

3 Concluding RemarksSummary & Conclusion

4 Appendix: Supplementary Examples (Non-identical, Non-generic)When The Power Condition FailsWhen The Cost Condition Fails

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All-Pay Contests

We study a class of games, “all-pay contests,” which capture generalasymmetries and sunk/irreversible investments inherent in manyscenarios:

I LobbyingI Competition for market powerI Labor-market tournamentsI R&D races

We provide a closed-form formula for players’ equilibrium payoffsand analyze player participation.

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Each player chooses a costly score and can be in one of two states:winning or losing.The primitives of the contest are commonly known: the equilibriumpayoffs represent economic rents (not information rents).Our cost functions allow for differing production technologies, costs ofcapital, and prior investments. Nonordered or state-dependent costfunctions are also allowed.Separable contests

I When all investments are unconditional, each player is characterized byher valuation for a prize (the payoff difference between the twostates) and a weakly increasing, continuous cost function thatdetermines her cost of choosing a score independently of the state.

I It nests many models of competition that assume a deterministicrelation between effort and prize allocation.

I Single- and multi-prize all-pay auctions are separable contests withlinear costs.

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An Example

Three risk-neutral firms compete for a monopoly position (v = 1).Each firm chooses how much to invest in lobbying (=score).

Firms 1 and 2 have better lobbying technologies.Firm 3 has an initial advantage, yet her cost for high scores is high.1− K (with score 1+ ε) is a lower bound on 1’s expected payoff.It can be better off: firm 1 must employ a mixed strategy in anyequilibrium.

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An Example - What Our Results Imply

Firm 1’s equilibrium payoff is 1− K . Firms 2 and 3 can guaranteethemselves (no more than) 0 (Theorem 1).Precisely one player receives a strictly positive expected payoff.For low and strictly positive values of γ, all three players mustparticipate (↔Theorem 2). This participation results from thenonordered nature of players’ cost functions.Lowering the prize’s value can lead to a positive payoff for player 3,making her the only player who obtains a positive expected payoff.

A VARIANT: What if firm 3 has 0 marginal cost?I Firm 3 wins with certainty.I It may be that no player invests even if a contest provides a valuable

prize (pure-strategy equilibrium).

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The Model

n players compete for m homogenous prizes (0 < m < n).The set of players: N = {1, ..., n}.Players compete by each choosing a score simultaneously andindependently.

I Player i chooses a score si ∈ Si = [ai ,∞),where ai ≥ 0 is her initial score.

Each of the m players with the highest scores wins one prize.

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Player i has preferences over lotteries w/ an outcome pair (si ,Wi ):

Wivi (si )− (1−Wi )ci (si )

where vi : Si → R is i ’s valuation for winning (ci : cost of losing).Given a profile of scores s = (s1,..., sn), player i ’s payoff is

ui (s) = Pi (s)vi (si )− (1− Pi (s))ci (si )

where Pi : ×j∈NSj → [0, 1], player i ’s probability of winning s.t.

Pi (s)

0 if sj > si for m or more players j 6= i

1 if si > sj for N −mor more players j 6= i

any value in [0, 1] otherwise

s.t.∑n

j=1 Pj(s) = m.A player’s probability of winning depends on all players’ scores.Her valuation for winning and cost of losing depend only on herchosen core.

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Three Assumptions

Assumption 1: vi and −ci are continuous and non-increasing.

Conditional on winning or losing, a lower score is weakly preferable.

Assumption 2: vi (ai ) > 0 and limsi→∞vi (si ) < ci (ai ) = 0.

With the initial score, winning is better than losing, so prizes are valuable.However, losing with the initial score is preferable to winning withsufficiently high scores.

Assumption 3: ci (si ) > 0 if vi (si ) = 0.

If winning with score si is as good as losing with the initial score, thenwinning with score si is strictly better than losing with score si :vi (si ) = ci (ai ) = 0 > −ci (si )

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This formulation allows the difference between a player’s valuation forwinning and her cost of losing to depend on her chosen score.

Consider a competition for promotions in which the value of the prizefor player i is fixed at Vi but some costs are only borne if the playerwins (cWi when she wins and cLi when she loses). Then:

ui (s) = Pi (s)(Vi − cWi (si )

)− (1− P(s)) cLi (si )

Contests can capture players’ risk attitudes as well. Let

vi (si ) = f (1− P(s))

andci = f (cLi )

for some strictly increasing f s.t. f (0) = 0.

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Separable Contests

Every player i ’s preferences over lotteries with outcomes (si ,Wi )depend only on the marginal distributions of the lotteries.The effect of winning or losing on a player’s Bernoulli utility isadditively separable from that of the score:

vi (si ) = Vi − ci (si )

andui (s) = Pi (s)Vi − ci (si )

for Vi = vi (ai ) > 0.I The value ci (si ) could be thought of as player i ’s cost of choosing score

si .I Vi could be thought of as player i ’s valuation for a prize.I All expenditures are unconditional, and players are risk neutral.

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Key Concepts

1 Player i ’s reach ri is the highest score at which her valuation forwinning is 0:

ri = max{si ∈ Si |vi (si ) = 0}2 Player m + 1 is the marginal player.3 The threshold T of the contest is the reach of the marginal player:

T = rm+1

4 Player i ’s power wi is her valuation for winning at the threshold:

wi = vi (max{ai ,T})

In a separable contest, a player’s reach is the highest score she canchoose by expending no more than her valuation for a prize.

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Generic ContestsGeneric Conditions

1 Power Condition:The marginal player is the only player with power 0.

2 Cost Condition:The marginal player’s valuation for winning is strictly decreasing at thethreshold. For every x ∈ [am+1,T ),

vm+1(x) > vm+1(T ) = 0

In a separable contest, in particular, for every x ∈ [am+1,T ):

cm+1(x) < cm+1(T ) = Vm+1

Players in NW = {1, ...,m} have strictly positive powers, and playersin NL = {m + 1, ..., n} have strictly nonpositive powers.Contests that do not meet the Generic Conditions can be perturbedslightly to meet them.

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Payoff ResultTheorem 1In any equilibrium of a generic contest, the expected payoff of every playerequals the maximum of her power and 0.

Players in NW have strictly positive expected payoffs (NL has 0).A generic contest has the same payoffs in all equilibria.

ProofWe invoke Least Lemma, Tie Lemma, Zero Lemma, and Threshold Lemmato prove Theorem 1.

A mixed strategy Gi of i is a cumulative probability distribution thatassigns probability 1 to her set of pure strategies Si .When a strategy profile G = (G1, ...,Gn) is specified, Pi (x) isshorthand for player i ’s probability of winning when she choosesx ≥ ai with certainty and all other players play according to G.For an equilibrium (G1, ...,Gn), denote by ui = ui (Gi ) player i ’sequilibrium payoff.

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Step 1: Least LemmaA player’s expected payoff in G is at least the maximum of her power and 0.

It suffices to consider players with strictly positive power (NW ).In equilibrium, no player chooses scores higher than her reach with astrictly positive probability (∵A1, A3).By choosing max{ai ,T + ε} for ε > 0, a player i in NW beats allN −m players in NL with certainty.For every player i in NW , by continuity of vi ,

ui ≥ vi (max{ai ,T + ε})→ε→0 vi (max{ai ,T}) = wi

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Step 2: Tie LemmaSuppose that in G two or more players have an atom at a score x . Thenplayers who have an atom at x either all win with certainty or all lose withcertainty when choosing x .

N ′: the set of players who have an atom at x : |N| ≥ 2.E : strictly positive-probability event that all players in N ′ choose

x .D ⊂ E : the event in which a relevant tie occurs at x ; i.e. the event in

which m′ prizes are divided among the |N ′| players in N ′,with 1 ≤ m′ < |N ′|.

Conditional on D > 0, at least one player i in N ′ can strictly increaseher probability of winning to 1 by choosing x + ε. Since i chooses xwith strictly positive probability, x ≤ ri and vi (x) > −ci (x)(∵ A1− A3).

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By continuity of vi and ci , player i would be strictly better off bychoosing a score x + ε ⇒ D has probability 0 and

P(E ) = P(EL) + P(EW )

EL ⊂ E is the event that at least m players in N \ N ′ choose scoresstrictly higher than x .

EW ⊂ E is the event that at most m − |N ′| players in N \ N ′ choosescores strictly higher than x .

Either EL or EW have probability 0 by independence of players’strategies (otherwise, D would have strictly positive probability).

1. If P(E ) = P(EL), then, without conditioning on E , at leastm players in N \ N ′ choose scores strictly higher than x withprobability 1, so Pi (x) = 0 for every player i in N ′.

2. Similarly, if P(E ) = P(EW ), then Pi (x) = 1 for every playeri in N ′.

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Step 3: Zero LemmaIn G, at least n −m players have best responses with which they win withprobability 0 or arbitrarily close to 0. These players have an expected payoffof at most 0.

J: a set of some m + 1 playersS̃ : the union of the best-response sets of the players in J

sinf : the infimum of S̃

Case 1: two or more players in J have an atom at sinf :N ′ ⊂ J: the set of such players in J.It cannot be that Pi (sinf ) = 1 for every player i in N ′, because anyplayer in J \ N ′ chooses scores strictly higher than sinf with probability1, even if the players in N \ J choose scores strictly lower than sinfwith probability 1, because only

m −∣∣(J \ N ′)∣∣ = m − (m + 1−

∣∣N ′∣∣) = ∣∣N ′∣∣− 1 > 0

prizes are divided among the |N ′| players in N ′.Thus, by Tie Lemma, Pi (sinf ) = 0 for every player i in N ′.

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Case 2: One player (= i) in J has an atom at sinf :Pi (sinf ) = 0, since all m players in J \ {i} chooses scores strictlyhigher than sinf with probability 1.

In cases 1 and 2, Pi (sinf ) = 0 for some player i in J who has an atom atsinf , so sinf is a best response for this player at which she wins withprobability 0.

Case 3: No player in J has an atom at sinf .By definition of sinf , there exists a player i in J with best responses{xn}∞n=1 → sinf .Since 1 ≥ 1− Pi (xn) ≥

∏j∈J\{i}(1− Gj(xn)), no player in J has an

atom at sinf , and G is right-continuous:

Pi (xn)→ 0 as n→∞.

Note that J was a set of any m + 1 players. That is, even if J includes allthe winning players, there are at least n − (m + 1) + 1 = n −m players inN who have best responses with which they win with probability 0.

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Comments on Three Lemmas

They hold regardless of the Generic Conditions.The Least Lemma + the Power Condition:

I The m players in NW have strictly positive expected payoffs.

The Least Lemma + the Zero Lemma:I Under the Power Condition, the n −m players in NL obtain expected

payoffs of 0.

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Step 4: Threshold LemmaThe players in NW have best responses that approach or exceed thethreshold and, therefore, have an expected payoff of at most their power.

Players in NL \ {m + 1} have strictly negative powers. Their reachesand the supremum of their best responses are strictly below thethreshold.There is some ssup < T s.t. Gi(x) = 1 for every player i inNL \ {m + 1} and every score x > ssup.This implies that every player i in NW chooses scores that approach orexceed the threshold (i.e. has Gi (x) < 1 for every x < T ).Otherwise, for some s in (ssup,T ), Gi (s) = 1 for all but at mostm − 1 players in N \ {m + 1}.But then the marginal player could win with certainty by choosing ascore in (max{am+1, s},T ) (NOTE: am+1 < T ).This would give her a strictly positive payoff, a contradiction.

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Take a player i in NW . Because Gi (x) < 1 for every x < T , thereexists a sequence {xn}∞n=1 of best responses for player i that approachsome zi ≥ T .Since xn is a best response for player i , who has a strictly positivepayoff by the Least Lemma and the Power Condition, vi (xn) > 0.By A1 and A2, vi (xn) > −ci (xn). By continuity of vi , we have

ui = ui (xn) = Pi (xn)vi (xn)− (1− Pi (xn)) ci (xn) ≤ vi (xn)

→xn→zi vi (zi ) ≤ vi (T ) = wi

(Proof of Theorem 1) The Least Lemma + the Threshold Lemmashow that players in NW have expected payoffs equal to their power.We checked that players in NL have expected payoffs of 0. SinceNL ∪ NW = N, the expected payoff of every player equals themaximum of her power and 0.

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Discussion of the Payoff Characterization

Equilibrium payoffs in generic contests depend only on players’valuations for winning at the threshold.

I Only the reach of each player and valuations for winning at thethreshold need to be computed.

I Players’ costs of losing do not affect payoffs.

The number of players who obtain positive expected payoffs equals thenumber of prizes.

Simon and Zame’s (1990) show that an equilibrium exists for sometie-breaking rule.The following corollary of their result and the Tie Lemma above showthat an equilibrium exists for any tie-breaking rule.

Corollary 1Every contest has a Nash equilibrium.

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Contests That Are Non Generic

Corollary 2Every contest (generic or not) has at least one equilibrium in which everyplayer’s payoff is the maximum of her power and 0.

Proof Sketch This can be proved by considering a sequence of genericcontests that approach the original contest and anequilibrium for each contest in the sequence.Every limit point of the resulting sequence of equilibria is anequilibrium of the original contest in which payoffs are givenby the payoff result.

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Corollary 3In any equilibrium of a contest in which all players are identical, all playershave a payoff of 0.

Proof Sketch In any equilibrium of any contest, identical players haveidentical payoffs, and the Zero Lemma shows that at leastone player has payoff 0.

When players are not identical and the contest is not generic, thepayoff of a player in some equilibrium may be very close to hervaluation for winning at initial score, even if her power is very low.

I See Example 1 and Example 2.

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Participation

Theorem 2In a generic contest, if the normalized costs of losing for the marginalplayer are strictly lower than that of player i > m + 1, that is,

cm+1(max{am+1, x}vm+1(am+1)

<ci (x)

vi (ai )for all x ∈ Si s.t. ci (x) > 0,

and the normalized valuations for winning for the marginal player areweakly higher than that of player i > m + 1, that is,

vm+1(max{am+1, x}vm+1(am+1)

≥ vi (x)

vi (ai )for all x ∈ Si ,

then player i does not participate in any equilibrium. In particular, if theseconditions hold for all players in NL \ {m + 1}, then in any equilibrium onlythe m + 1 players in NW ∪ {m + 1} may participate.

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Comment on Theorem 2

When players’ costs are strictly ordered, at most m + 1 playersparticipate in any equilibrium.However, a player in NL \ {m + 1} may participate if she has a localadvantage w.r.t. the marginal player.

I In our first example, suppose player 3 did not participate.I Players 1 and 2 play strategies that make all scores in (0,T ) best

responses for both of them.I For low values of γ > 0, player 3 could then obtain a strictly positive

payoff by choosing a low score: a contradiction.I Thus, player 3 must participate in any equilibrium, even though her

expected equilibrium payoff is 0.

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Proof of Theorem 2 (in Appendix)

It suffices to prove the result for contests in which vi (ai ) = 1 ∀i .Choose an equilibrium G of such a contest.Suppose player i > m + 1 that meets the conditions of theproposition participated in G.Let ti = inf {x : Gi (x) = 1} < T and t̃i = max{am+1, ti}. Then,

I t̃i < TI Pi (ti ) < 1 (Pf. of the Threshold Lemma)I For every δ > 0, Pm+1(t̃i + δ) ≥ Pi (ti ) since by choosing (t̃i + δ),

player m + 1 beats player i for sure.

For every δ > 0 s.t. t̃i + δ < rm+1 = T , we have

vm+1(t̃i + δ) > 0 ≥ −cm+1(t̃i + δ)

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Therefore,

um+1 ≥ Pm+1(t̃i + δ)vm+1(t̃i + δ)− (1− Pm+1(t̃i + δ))cm+1(t̃i + δ)

≥ Pi (ti )vm+1(t̃i + δ)− (1− Pi (ti ))cm+1(t̃i + δ)

By definition of participation, ci (ti ) > 0, so ci (ti ) > cm+1(t̃i ).Since Pi (ti ) < 1 and vm+1(max{am+1, x}) ≥ vi (x) for all x ∈ Si , bycontinuity of vm+1 and cm+1, player m + 1 can choose t̃i + δ forsufficiently small δ > 0 s.t.

Pi (ti )vm+1(t̃i + δ)− (1− Pi (ti ))cm+1(t̃i + δ)

> Pi (ti )vi (ti )− (1− Pi (ti ))ci (ti )

= ui (ti )

⇒ um+1 > 0, which contradicts the payoff result (wm+1 = 0).

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Summary & Conclusion

All-pay contests capture general asymmetries among contestants andallow for both sunk and conditional investments.

Thm 1: Expected PayoffsThe expected payoff of everyplayer equals the maximum ofher power and 0; i.e.max {wi = vi (max{ai ,T}), 0}

Thm 2: ParticipationPlayers that are disadvantagedeverywhere w.r.t. the marginalplayer do not participate in anyequilibrium

Reach and power are the right variables to focus.The addition of a player makes existing players weakly worse off.The addition of a prize makes player m + 2 the marginal player.This lowers the threshold and makes existing players better off.Making prizes more valuable raises the threshold.

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Example 1: The Power Condition FailsThe Payoff Result Does Not Hold

One prize of common value 1Players’ costs are:

c1(x) =

{(1− α)x , if 0 ≤ x ≤ h

(1− α)h + (1+ αh1−h )(x − h), if x > h

c2(x) =

{(1− ε)x , if 0 ≤ x ≤ h

(1− ε)h + (1+ εh1−h )(x − h), if x > h

c3(x) =

{γx , if 0 ≤ x ≤ h

γh + L(x − h), if x > h

for some small α, ε in (0, 1), small γ ≥ 0, h in (0, 1), and L > 0.Regardless of the value of L, the threshold is 1 and the PowerCondition is violated (∵ at least two players have power 0).The Cost Condition is met (costs are strictly increasing at 1).

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For any h in (0, 1), there exist some β > 0 and M > 0 s.t. ifα, ε, γ < β and L > M, then (G1,G2,G3) is an equilibrium, for:

G1(x) =

0, if x < 0(1− ε)h + γ

1−α(xh − 1), if 0 ≤ x ≤ h

(1− ε)h + (1+ εh1−h )(x − h), if h < x ≤ 1

1, if x > 1

G2(x) =

0, if x < 0(1− α)h, if 0 ≤ x ≤ h

(1− α)h + (1+ αh1−h )(x − h), if h < x ≤ 1

1, if x > 1

G3(x) =

{x/h, if x ≤ h

1, if x > h

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Players’ costs

Players’ atoms and densities in the equilibrium

As L increases, player 3’s power, 1− γh − L(1− h), becomesarbitrarily low. As h→ 1 and ε, α, γ → 0, for any value of L > M,player 3 wins with near certainty. Her payoff approaches the value ofthe prize: (1− ε)(1− α)h2 − γh→ 0.A slight change in player 1’s or 2’s valuation for the prize leads to ageneric contest and destroys the equilibrium.

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Example 2: The Cost Condition FailsThe Payoff Result Does Not Hold

Competition may stop before the threshold is reached.Two-player separable contest for one prize of common value 1:

c1(x) = bx , for some b < 1

c2(x) =

xd , if 0 ≤ x < d

1, if d ≤ x ≤ 12x − 1, if x > 1

for some d in (0, 1).

Reach (ri ) Power (wi )

Player 1 1b > 1 1− b > 0

Player 2 1 0

The Power Condition holdsThe Cost Condition fails: c−1

2 (c2(r2)) = [d , 1].43 / 45

(G1,G2) is an equilibrium in which player 1 has a payoff of1− bd > w1 for

G1(x) =

0, if x < 0xd , if 0 ≤ x ≤ d

1, if x > d

G2(x) =

0, if x < 01− bd − bx , if 0 ≤ x ≤ d

1, if x > d

As b → 1, w1 → 0However, for any value of b, as d → 0, player 1’s payoff approaches 1(=the value of the prize).

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Example 2: Another Equilibrium

(G1, G̃2) is an equilibrium in which both players’ payoffs equal theirpowers, for

G̃2(x) =

0, if x < 01− b + bx , if 0 ≤ x ≤ 11, if x > 1

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all-paycontestsfaculty.haas.berkeley.edu/rjmorgan/phdba279b/phdba297b_4...1 introduction...

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