Parham Holakouee
PHDBA 279B
Spring 2012
Contests
Strategy in Contests – An Introduction(Kai Konrad, 2007)
I. Basic Setup for Contests
•Contestants N = {1, …, n}
•Each Contestant (“C”), i, makes effort xi
•Vector X = {x1…,xn} of efforts
•Prize allocated to Contestant(s)
•Simplest Case: One Contestant receives prize = B, others get zero.
•Contest Success Function:
•Vector of efforts into Pr of i winning: pi = pi(x1, …, xn)
•pi’s are between 0 to 1, sum to 1 (unless Pr prize not allocated)
•Cost of Effort: Ci(xi)
•Simplest assumption is that cost of effort is equal to effort exerted: Ci(xi) = xi
•Payoff to i: πi(x1…,xn) = pi(x1…,xn)vi(B) – Ci(xi)
II. Real-world Examples of Contests
•Advertising Campaigns
•Major share of these efforts are made up-front, prior to actual sales.
•Best understood as effort choices determining market shares.
•Contest success function does not have a probabilistic interpretation in this case but as share in total market.
•Public Good: advertising could potentially increase the overall sizeof the market and thus carries properties of a public good.
•Litigation
•Fee-shifting rules play an important role in optimal effort levels.
•Difference between American (each litigant pays own cost) and English rule (loser pays both).
•Labor Market Tournaments
•Tournament could serve two functions:
•(a) a reward scheme to induce worker effort, or
•(b) to identify employees who are particularly skilled at superior tasks or at assuming more responsibility.
•Relative rewards may induce destructive effort intended to reduce performance of competitors.
•Tournament structure to reduce incentives tosabotage.
•R&D
•Rents to firm who introduces new product first – with no competitors offering same product.
•Politics
•Analogous to advertising campaigns. Distinguished from firm marketing since politicians benefit from a larger market and payoff is not necessarily a smooth function of market size and politician’s respective market share.
•Military
•“Effort” can be amount of country’s resources devoted to building military capability versus basic consumption goods for citizenry.
•Sports
•Effort typically not recoverable after outcome.
•Non-monetary benefits of prize structure critical to effort strategy.
III. We will distinguish between three basic types of contests:
A. Type 1: First-Price All-Pay Auction(FPAPA)(Hillman and Riley, 1989)
- Setup
•Contestants: i = 1, 2
•Prize Value: v1 ≥ v2 ≥ 0
•Complete Info
•Valuation is Non-negative
•Concurrent Effort Decision: xi≥ 0
•Effort levels are unequivocally observed
•Player exerting higher effort always wins prize
•No Noise
•C(xi) = xi
•Probability C1 wins: p1(x1, x2) =
•Pr = 1 if x1x2
•Pr = ½ if x1 = x2
•Pr = 0 if x1x2
- Optimal Effort Levels in the FPAPA with Different Valuations
•Optimal Strategy for C1:
•(a) x1= x2+ ε or
•(b) x1= 0
•No Pure Strategy Equilibrium
•If x1is the optimal reply to x2, x2 can’t be optimal reply to x1.
•C2 will never exceed x2 = v2
•C1 can win for certain with x1 = v2 + ε
•v2 = Upper limit of reasonable efforts
•Problem reduced to C’s randomizing on [0, v2]
- Equilibrium Payoffs
•C1 = v1 – v2
•C2 = 0
•Payoffs in mixed strategy must be equal to Equil Payoffs
•Mixed Strategy CDFs:
F1(x1) = x1/v2forx1∈ [0, v2]
1forx1v2
F2(x2) = [1-v2/v1] + x2/v1forx2∈ [0, v2]
1forx2v2
- Mixed Strategy CDFs are Optimal Replies
•C1’s Payoffs = v1 – v2for allx1∈ [0, v2] when playing against C2 randomizing according to CDF F2(x2).
•C2’s Payoffs = 0 for allx2∈ [0, v2] when playing against C1 randomizing according to CDF F1(x1).
•π1(x1) = F2(x1)v1 – x1 = v1 – v2
•= [1 – v2/v1]v1 + (x1/v1)v1 – x1
•= v1 – v2 + x1 – x1
•= v1 – v2
•π2(x2) = F1(x2)v2 – x2 = 0
•= (x2/v2)v2 – x2
•=x2– x2
•= 0
- Expected Efforts
•E(x1) = (v2)/2
•E(x2) = (v2)^2/2(v1)
•Sum of expected efforts falls short of v2; v2 = effort in a standard 2nd Price Auction.
•Prize not necessarily allocated to C with highest valuation.
•This “inefficiency” disappears if players move sequentially.
B.FPAPA with More Than 2 Players(Baye, Kovenock, deVries, 1996)
•N ≥ 2
•Ex. 1
•v1 ≥ v2 ≥ v3 ≥ … ≥ vn
•xi = 0 for all i > 2
•C1 and C2 choose CDFs as in n=2 case
•Ex. 2
•v1 = v2 = … = vj > vj+1 ≥ … ≥ vn
•xi = 0 for all i > j
•Full continuum of equilibria:
•Any number between 2 and j highest bidders may make positive bids.
•If only 2 players, bid according to CDF of n=2.
C. FPAPA: Cost Variants(Baye, Kovenock, deVries, 1998)
•v – bx1– dx2if C1 wins
•-ax1 – tx2if C1 loses
•Standard Contest: b = a =1 and d = t = 0
•British Legal System: b = d = 0 and a = t = 1
•Loser pays all
•War of Attrition: b = t = 0 and d = a = 1
•Both pay loser’s costs
D. FPAPA: Constraints(Che and Gale, 1998)
•n contestants, limit = m
•If m < (1/n) min {v1, v2,…, vn}
•All will choose max effort = m
•Lottery with n-1 other C’s
•m < (1/k) min {v1,…, vk}
•C1 through Ck will participate
•Eg. v1 and v2 participate when: m < (1/2) min {v1, v2}
•If mv2, effort limit non-binding in Eq
•If m between [½min{v1, v2}, min{v1, v2}]
•Each C chooses effort: (0, 2m – v2] ∪ {m}
•C1’s payoff = v1 – v2
•C2’s payoff = 0
E. FPAPA: Incumbency(Baye, Kovenock, deVries, 1998)
•Ci(x) = x, and v1 = v2 = v
•But C1 has ‘headstart advantage’ = δ
•Probability C1 wins: p1(x1, x2):
•Pr = 1 if x1x2 - δ
•Pr = ½ if x1 = x2 - δ
•Pr = 0 if x1x2 – δ
•CDFs:
F1(x1) = δ/v + (x1)/vforx1∈ (0, v-δ)
1forx1 ≥ v-δ
F2(x2) = δ/v for x2 ∈ (0, δ)
x2/vforx2∈ [δ, v)
1forx2 ≥ v
F. FPAPA: One-sided Asymmetric Info (Shogren, 1998)
•Valuations drawn from distribution vi∈ [a,b], a < b
•Qa = Pr valuation at a
•Qb = Pr valuation at b = (1 – Qa)
•v1: public, v2: private
•For C1, CDF different for v1 = a, v1 = b
•For C2, CDF depends on both C’s type
•Expected Eq Payoff C2 = (b-a)QaQb(2 – Qa(b-a / b))
•C1 = QaQb(b-a)
•C2 gets Info Rent
IV. Type 2: Additive Noise (Lazear and Rosen, 1981)
•Effort xi does not deterministically yield what’s observed
•Outcome of contest not unequivocally contingent on X = {x1,…,xn} of efforts.
•xi (effort) + εi (noise)
•ε = ε2 – ε1
•Pr = 1 if x1 –x2 > ε
•Pr = ½ ifx1 – x2 = ε
•Pr = 0 if x1 - x2< ε
•G(ε) is the distribution of ε from [-e, e], e > 0
•Winning Prize = Bw
•Losing Prize = Bl
•C’s Risk Neutral
•C’s maximize:
•pi(x1, x2)(vi(Bw) – vi(Bl)) – C(xi) + vi(Bl)
•C(xi) no longer linear:
•Marginal effort cost is increasing in the probability of winning.
•Convex function of effort.
•C(0) = 0, C’(xi) > 0, C’’(xi) > 0
•F.O.C.: ∂G(x1 – x2)/∂x1[v1(Bw) – v1(Bl)] = C’(x1)
•Analogous for C2
•Simplifying Assumptions:
•(1) vi(b) = b and same cost function; x1 = x2
•F.O.C. simplifies to: G’(0) (Bw) – (Bl) = C’(xi)
•(2) ε uniform on [-e, e]; FOCs independent
•∂G(x1 – x2)/∂x1 = 1/2e
•From F.O.C., effort is reduced as:
• (1) ε is more disperse, and
• (2) Bw is reduced
•Organizer will choose Bw – Bl such that x* = x1* = x2*:
•G’(0)(Bw – Bl) = C’(x*) also fulfills
•C’(x*) = ζ’(2x*)
•[ζ = Organizer’s benefit from C’s efforts x1 + x2]
V. Type 3: Tullock Contest(Tullock, 1980)
•pi(x1,…,xn) = xi^r/Σjj=1 to n xj^rif max{x1,…,xn} > 0
1/notherwise
•Pr of winning is ratio of own effort over the sum of total efforts.
•r = 1: ‘Lottery Contest’
•As r approaches Infinity, Contest Function w/out noise.
•Ci(xi) = xi
•Existence:
•Pure Strategy Nash Eq exists iff r ≤ n/n-1
•If r > n/n-1, typically mixed Nash Eq
•Σ(xi) > B is possible, since there is mixing
•But Expectation of (Σxi) ≤ B
•2 players, F.O.C.: rxi^r-1 xj^r/xi^r + xj^r (v1) = 1
•xi = (r(n-1)/n^2)(B)
•Σxi = (r(n-1)/n)(B)
•With r = 1, rents dissipate as n increases.
•As r increases, players will choose not to participate if this entails negative expected rents.
•Thus, expected equilibrium payoffs to Players will not drop below zero.
VI. Experimental Evidence and Evolutionary Game Theory
•In experiments, effort levels exceed NE predictions.
•Greater over-dissipation in Tullock than FPAPA.
•Absolute Fitness vs. Relative Fitness
•Absolute Material Payoff: πi = xi – xi
•πi(α) = απi + (1-α)[πi – 1/n-1 Σj≠iπj]
•α = 1: Absolute Performance matters only
•α = 0: Relative Performance matters only
•Disposition for enjoying relative rewards could be hard-wired in our biology
•Indirect Approach in Evolutionary Economics
•Pre-determined effort levels sub-optimal when unforeseen changes or cyclical fluctuations.
•Lends support to a more complex evolutionary rule.
VII. Extensions
Extensions: Timing(Dixit, 1987)
•Endogenous Timing:
• r=1, v1 > v2
•Each C can move e = early or l = late
•Asymmetry in valuations, leads to endogenous asymmetry in timing preference.
•C2 will move first and becomes Stackelberg leader.
•C1 waits and observes C2’s effort before choosing effort to max payoff.
•This asymmetry in timing reduces aggregate effort.
•Not fully robust, but holds for all-pay without noise.
•Commitment is critical.
•Optimal strategy for first-mover will be to change original effort after second player has moved.
Extensions: Exclusion(Baye, Kovenock, deVries, 1997)
•Not always optimal to admit C who values Prize most highly.
•Tradeoff between high valuation of C’s and contest homogeneity.
•v1v2 = v3, Ci(xi) = xi
•One of 2 equilibria with highest effort (n=2, all-pay, v1, v2):
•v2/2(1 + v2/v1) < v2
•But if exclude C1, Eq again (n=2, all-pay, v2, v3)
•v2/2 + v2/2 = v2
•Asymmetry results in discouragement of weakest C (zero effort w/ high Pr).
•Weak C influences effort of strong C.
•Homogeneity is critical for inducing effort exertion (especially for all-pay, no noise).
•Tullock and FPAPA with noise are less sensitive to value homogeneity.
Extensions: Prize Structure(Moldavanu and Sela, 2002)
•Ci(xi) = xi, but different budgets: w1 = 10, and w2 = w3 = w4 = 1
•Prize money b = 2
•With 1 prize = 2, one equilibria has 2 players making positive bids:
•Ex(x1) + Ex(x2) = 1
•But if 2 prizes = 1, then 2 simultaneous Eq
•Budget constraints now are not binding, and competition is symmetric.
•Ex(x1) + Ex(x2) + Ex(x3) + Ex(x4) = 2
Robust Results
•Increased effort of one C is a negative externality for all others.
•For a given Prize, C can generally increase own rents if aggregate effort of all C’s reduced.
•More heterogeneousC’s, reduces aggregate efforts and increases aggregate net payoffs.
•C with highest valuation exerts highest effort and has highest probability of winning.