Dividing the Spoils in Team Contests

Technical Summary: Dividing the Spoils in Team Contests

Problem Statement
This paper investigates the strategic allocation of a collective prize within a team competing in a multi-battle contest. The setting involves two rival teams (A and B), each managed by a decision-maker who must simultaneously partition a team-specific budget ($W_i$) among $2N+1$ heterogeneous pairwise battles. The team that wins a majority of battles claims the collective prize. Unlike standard contest models where prize structures are exogenous, this study endogenizes the internal prize division. Managers choose prize shares ($v_{it}$) for each battle $t$ conditional on team victory to maximize the probability of winning the majority. The core strategic challenge is that a player's effort in a specific battle depends not only on the direct opponent's effort but also on the "pivotality" of that battle—how likely a win in that specific front is to determine the overall team victory. Furthermore, the manager's objective function is highly nonlinear, as the probability of winning a majority is not quasiconcave in the vector of battle-winning probabilities, complicating the existence of equilibrium.

Methodology
The authors model the interaction as a two-stage game:

1. Stage 1: Managers simultaneously announce prize allocation vectors $v_i = (v_{i1}, \dots, v_{i,2N+1})$ subject to budget constraints $\sum v_{it} = W_i$.
2. Stage 2: Matched players in each battle exert effort $e_{it}$ according to a generalized Tullock contest success function (CSF) with discriminatory power $r_t \in (0, 1]$. The probability of winning battle $t$ is determined by the ratio of effective efforts.

The analysis proceeds by solving the second stage to derive reduced-form winning probabilities, which are shown to depend only on the ratio of prizes and costs within that specific battle, provided no team has already secured a majority. The first stage is then analyzed using game-theoretic tools. A critical technical hurdle is establishing the existence of a pure-strategy Nash equilibrium. The authors demonstrate that the team's winning probability is log-concave in the manager's allocation strategy. This property is non-trivial because the majority aggregation function is not quasiconcave. The proof of log-concavity relies on showing that a specific structured matrix, derived from the conditional covariance of battle outcomes given team victory, is positive semidefinite. This is achieved through a coefficientwise decomposition of a polynomial matrix into blocks that are verified to be positive semidefinite.

Key Results
The paper characterizes a unique pure-strategy subgame-perfect Nash equilibrium with the following properties:

1. Proportional-Prize Property: In equilibrium, both managers allocate their budgets across battles in identical relative proportions. Specifically, the ratio of prizes assigned to any battle $t$ by the two teams equals the ratio of their total budgets:
   $$\frac{v^*_{Bt}}{v^*_{At}} = \frac{W_B}{W_A}$$
   This holds regardless of heterogeneity in player costs ($c_{it}$) or battle technologies ($r_t$). If budgets are equal ($W_A = W_B$), the absolute prize shares are identical ($v^*_{At} = v^*_{Bt}$).

2. Salience-Based Allocation: The common share of the budget assigned to battle $t$ is proportional to its salience ($S_t$):
   $$v^*_{it} = W_i \cdot \frac{S_t}{\sum_{\tau} S_\tau}$$
   Salience is defined as the product of three components:

   • Discriminatory Power ($r_t$): The sensitivity of the battle outcome to effort.
   • Symmetry ($p^*_{At} p^*_{Bt}$): The product of the equilibrium winning probabilities, which is maximized when players are evenly matched.
   • Pivotality ($\theta^*(t)$): The probability that the remaining battles split evenly, making battle $t$ decisive for the majority.

   Consequently, resources flow to battles that are simultaneously contestable (high symmetry and discriminatory power) and consequential (high pivotality). Near-predetermined battles or those unlikely to affect the majority outcome receive minimal shares.

3. Robustness to Temporal Structure: The equilibrium allocation and winning probabilities are invariant to the temporal order of battles (simultaneous, sequential, or hybrid). This is because the marginal incentive for a player is the assigned prize multiplied by the increase in team-winning probability, a term that is identical for both competing players regardless of when the battle occurs.

4. Comparative Statics:

   • Strengthening a team (lowering member costs or increasing the team prize) strictly increases that team's winning probability, even as allocations adjust across all fronts.
   • In well-matched battles, allocation depends solely on the contest technology; more discriminating battles command larger shares.
   • Total aggregate effort cost is decomposed into the total budget, total salience, and the Herfindahl-Hirschman Index (HHI) of salience shares. Higher concentration of salience (inequality in battle importance) increases total effort costs.

Significance and Contributions
The paper makes three primary contributions to the literature on team contests and prize design:

1. Endogenizing Internal Allocation: While existing literature on team contests (e.g., Fu, Lu, and Pan, 2015; H¨afner, 2017) typically treats within-team prize sharing as exogenous, this paper solves the decentralized strategic problem where managers simultaneously design internal splits. This endogenization reveals that the "salience" of a battle is an equilibrium outcome, not a primitive.

2. Resolution of Existence in Non-Concave Environments: The paper provides a rigorous proof of equilibrium existence in a setting where the objective function (majority winning probability) is neither concave nor quasiconcave. By establishing log-concavity via a novel matrix decomposition technique involving conditional covariances, the authors extend the applicability of first-order conditions to global optimality in complex multi-battle environments.

3. The Proportional-Prize Convergence: The finding that rival managers converge on identical relative allocation rules, despite facing different cost structures and budget sizes, offers a strong theoretical prediction. It suggests that in majoritarian team contests, the relative distribution of resources is driven by the structural importance of the battles (pivotality and contestability) rather than the absolute capabilities of the teams. This contrasts with models where designers might bias allocations to favor weaker teams; here, the strategic interaction forces a convergence on the "most contested and most pivotal" fronts.

The results resonate with political economy literature on targeting swing voters (e.g., Str¨omberg, 2008; Dixit and Londregan, 1996) but differ by introducing strategic agents (players) on the battlefield, making contestability endogenous to effort rather than a fixed function of voter preferences. The framework is applicable to legislative elections, defense procurement, and team sports, offering a unified explanation for how organizations prioritize resources across multiple competitive fronts.