TheContest Gameis a simultaneous-move game of complete information characterized by two risk-neutral players 1 and 2, their common action space R+, and their payoff functionsU1,U2. The payoff functions and parameters are common knowledge.
Let i ∈ {1,2} represent a generic player and j ∈ {1,2} \ {i} her opponent. Players
i, j simultaneously choose ei,ej ≥ 0 levels of expenses. Player i values the prize at
vi > 0. A twice-continuously-differentiable success function θi : R2+ → [0,1] that satisfies additional assumptions to be set out below givesθi(ei,ej)as playeri’s probability of winning a prize according to her own belief; she believes that her opponent jwins the
prize with complementary probability 1−θi. Thus playeribelieves her and the opponent j’s respective probabilities of winning sum to 1.
Remark 13. It is possible to haveθ1+θ2 , 1. We interpret the scenario ofθ1+θ2 , 1
as capturing non-common beliefs regarding a "true" success function. Suppose there is a collection of success functions containingθ1, θ2. Before the players choose their expenses, Nature chooses one of these success functions to determine the outcome of the contest. At the time of choosing their respective expenses, both players know the collection of success functions, but they do not observe Nature’s choice. Player1assigns probability1to the event of Nature choosingθ1, while player2assigns probability1to Nature choosingθ2. See subsection 4.6.1 for an alternative scenario.
respectively characterize the winner’s spillovers and loser’s spillovers that player i’s
expenses ei generates to herself. Player i receives (self-generated) winner’s spillovers
wiieiif she wins, and loser’s spilloversliieiif she loses.
Playerialso may receive winner’s and loser’s spillovers generated by her opponent j’s
expensesej. Exogenous scalarswi j,li j ∈Randki > 0,i, j, characterize these spillovers.
Playerireceives winner’s spillovers wi jejki if she wins, and loser’s spilloversli jekji if she loses. The exponentkicharacterizes the returns ofejin generating spillovers to playeri.89 Example 1 contains a case whereki , 1. Callekji player j’s effective expenses affecting
player i, and ei player i’s effective expenses affecting herself. Thus player i receives self-generated, intra-player spillovers (wiieiandliiei) and opponent-generated, inter-player spillovers (wi jekji andli jekji).90
Playeri’spayoff isUi:R2+ →Rgiven by
Ui(ei,ej)=θi(ei,ej) h vi+wiiei+wi jekji i + 1−θi(ei,ej) h liiei+li jekji i −ei, (31) which is her expected outcome measured in monetary terms. She believes that if she wins, then she receives the sum of her prize value and her winner’s spillovers,vi+wiiei+
wi jekji. She believes that if she loses, then she receives the sum of her loser’s spillovers,
liiei+li jekji. Weightsθi(ei,ej), 1−θi(ei,ej)are respectively her probabilities of winning and losing according to her belief. Because she incurs expenses ei whether she wins or loses, the probability weights do not scale ei. This specification for the payoff function
assumes additive separability of expenses, spillovers and prizes. This means that expenses, spillovers and prizes are perfect substitutes and the players are risk neutral.
We now state and impose the following Assumptions 18-23 to guarantee equilibrium existence.
Assumption 18. Player i’s success function satisfies θi(ei,ej) = θi xkiei,xej
for any scalarx >0.
Assumption 18 requires player i to believe that the same ratio of effective expenses
— ei/e ki
j and x
kie
i/(xej)ki for x > 0 — leads to the same probabilities of winning.91 Intuitively, Assumption 18 requires ei and ej to be similarly effective in affecting prob-
89As subsection 4.6.2 will reveal, the present specification that each player’s expenses linearly produce
spillovers to herself captures the more general case of these spillovers being homogeneous functions.
90Sections 4.4 and 4.5 will utilize the distinction between these two notions of spillovers to capture
real-world scenarios.
abilities of winning and in producing spillovers. Because scalars 1 and ki respectively characterize the returns of ei and ej in generating spillovers to playeri, Assumption 18 ensures that 1 andkialso respectively characterize the returns ofeiandejin affecting her belief regarding her probability of winning; playeri’s belief regarding how expenses affect
her spillovers and probability of winning is thus consistent in this sense. Moreover, we aim to define a class of success functions that includes Tullock success functions, because they are the standard in contest theory (see section 4.1 and subsection 4.6.3). In the special case ofki =1, Assumption 18 reduces to the homogeneity of degree zero property, which is satisfied by Tullock success functions.
Assumption 19. Playeri’s success functionθi satisfies the following properties:
1. Her marginal probability of success with respect to her own expensesei is positive and non-increasing, and is bounded above. Formally, for a positive upper boundb¯
which may depend on her opponent’s expensesej, 0< ∂θi ∂ei ≤ b¯, ∂ 2θ i ∂ei2 ≤ 0.
2. Her marginal probability of success with respect to her opponent’s effective expenses
eki
j is non-decreasing, and is bounded below. Formally, for a lower boundbwhich may depend onei, ∂θi ∂eki j ≥ b, ∂ 2θ i ∂eki j 2 ≥ 0.
Together with Assumption 22 below, the upper and lower bounds that Assumption 19 impose prevent playerifrom have incentives to make unbounded expenses.
Definespillover differentialsδii, δi j ∈Rbyδii= wii−liiandδi j =wi j−li jrespectively. The value δiiei (respectively, δi jekji) is the difference between the winner’s and loser’s spillovers of player i arising from her own expenses ei (her opponent’s expenses ej). Rearrange playeri’s payoffUidefined by (31) to obtain
Ui(ei,ej)=θi(ei,ej)
h
vi+δiiei+δi jekji
i
− (1−lii)ei+li jekji. (32) Equation (32) expresses player i’s payoff as her expected benefits of winning — the
weighted sum of her prize value and spillover differentials,θi(ei,ej)
h
vi+δiiei+δi jekji
i
less herunweightedexpenses(1−lii)ei, and plus a termli jekji that does not vary with her own expensesei. The loser’s spilloversliieireduce her unweighted expenses becauseδiiei gives theadditionalspillovers of winning that her own expenses produce.
Assumption 20. Player i believes her spillovers are sufficiently small, in the following sense: δiiθi(ei,ej)+ h δiiei+δi jekji i∂θi ∂ei < 1−lii, (33)
where the inequality holds strictly in the limit whenei →0+.
Assumption 20 ensures that playeriwould have no incentives to make positive expenses
if the (non-spillover) prize of winning had zero value to her (vi = 0). Inequality (33) comes from taking the partial derivative of her payoffUiin equation (32) with respect to her own expensesei, and assumingvi =0. The left-hand side of inequality (33) sums her marginal (expected) benefits of making positive expenses, while the right-hand side her marginal unweighted expenses. Inequality (33) ensures her marginal unweighted expenses to be greater than her marginal benefits ifvi =0.
Assumption 21. Player i’s success function θi is sufficiently concave in the following sense: ∂2 ∂e2i h θ i(ei,ej) 1−lii−δiiθi(ei,ej) i ∂ ∂ei h θ i(ei,ej) 1−lii−δiiθi(ei,ej) i < 0.
Assumption 21 is a technical assumption that ensures player i’s payoff is strictly
quasiconcave in her own expenses.
Assumption 22. Playeribelieves that her success function and self-generated spillovers satisfy the following properties:
1. Her self-generated spillovers do not exceed her expenses. Formally,1 ≥ max{wii,lii}. 2. In the special case of her self-generated winner’s (respectively, loser’s) spillovers
covering her expenses, she does not win (lose) almost surely by making infinitely more (less) effective expenses than her opponent j does. Formally,
1= wii ⇒ lim
ei/ekij →+∞
1=lii ⇒ lim ei/ekij →0+
θi(ei,ej) >0.
Assumption 22 prevents playerifrom recouping all expenses by generating spillovers
alone. Part 1 bounds her marginal benefit of generating spillovers above by 1, her marginal cost of spending. In the special case where her spillovers cover her expenses if she wins (respectively, loses), part 2 prevents her from recovering all expenses almost surely by making infinitely more (less) effective expenses than her opponent does. Assumption 22 is trivially satisfied if her expenses strictly exceed her spillovers, 1> max{wii,lii}.
Assumption 23. The players’ expenses are insufficiently effective in affecting each other, in the sense ofk1k2 ≤ 2.
Assumption 23 restricts the effectiveness of the players’ expenses in affecting each other. Assumption 23 is trivially satisfied in the special case where each player’s expenses linearly affect her opponent, k1 = k2 = 1. In general, the product of k1 and k2 must be sufficiently small (in the sense of k1k2 ≤ 2) to satisfy Assumption 23. Intuitively,
Assumption 23 facilitates equilibrium existence by ruling out significant divergence in relative effective expenses. In all that follows, the previous assumptions hold.
The solution concept adopted is a Nash equilibrium that isnontrivialin the sense of comprising positive expenses by both players.