Additive multi-effort contests (Q2202231)

Let two players compete for a rent \(S\geq 0\). Each player \(i=1,2\) undertakes \(K_i\) efforts \(\vec{x_i} = (x_{ik})_{k=1}^{K_i}\) to win the rent; \(x_{ik}\geq 0\). The impact of effort \(x_{ik}\) is given by \(d_{ik}>0\). The decisiveness of effort \(x_{ik}\) is given by \(m_{ik}\); \(m_{ik} = 0\) -- no impact, \(0< m_{ik} <1\) -- less than proportional impact, \(m_{ik}=1\) -- proportional impact, \(m_{ik}>1\) -- more than proportional impact. The unit cost of effort \(x_{ik}\) is given by \(c_{ik}>0\). Finally, \(p_i(\vec{x_1}, \vec{x_2})\) stands for the probability that player \(i\) win the rent, and \(u_i(\vec{x_1},\vec{x_2})\) stands for the expected utility for player \(i\) when the multi-efforts \(\vec{x_2}\) and \(\vec{x_2}\) have been taken by player 1 and player 2, respectively. Denote by \(f_i(\vec{x_i})= \sum_{k=1}^{K_i} d_{ik}\cdot x_{ik}^{m_{ik}}\) the impact function of player \(i\) (production function with constant elasticity of substitution). Player \(i\) tries to increase the probability \(p_i(\vec{x_1},\vec{x_2}) = \frac{f_i(\vec{x_i})}{f_1(\vec{x_1})+ f_2(\vec{x_2})}\). Under its own multi-effort \(\vec{x_i}\) in the presence of the competitor multi-effort \(\vec{x_{3-i}}\). The total cost of \(i\)'s multi-effort is then \(C_i(\vec{x_i}) = \sum_{k=1}^{K_i} c_{ik}\cdot x_{ik}\). Hence the utility \(u_i(\vec{x_1}, \vec{x_2}) =p_i(\vec{x_1},\vec{x_2}) \cdot S - C_i(\vec{x_i})\). The innocuous looking form of \(f_i\) makes the analysis of the game with payoffs \(u_i:[0,\infty)^{K_1}\times [0,\infty)^{K_2} \to \mathbb{R}\) quite challenging. The analysis is based on the convexity and differentiation. Initially one observes that trivial parameters \(d_{ik}=0\) or \(m_{ik}=0\) yield Nash equilibria at the corners \(x_{ik}\), while nontrivial parameters allow for interior equilibria (all \(x_{ik}>0\)) under suitable circumstances. Furthermore, there appear several possibilities, two of which are the following extremes: (1) if \(0\leq m_{i1} \leq 1\) and \(0\leq m_{ik} <1\) for all \(2\leq k\leq K_i\), \(i=1,2\), then player \(i\) usually undertakes more than one effort; (2) if \(m_{ik}\geq 1\) for all \(1\leq k\leq K_i\), \(i=1,2\), then player \(i\) undertakes only one effort. The mixed case, \(m_{ik}\leq 1\) for \(k\leq K_{iq}\) and \(m_{ik}>1\) for \(k>K_{iq}\), where \(K_{iq}<K_i\), is more involved than (1) and (2). The rent dissipation and other vital questions, including the problem of coordination, are addressed in the article as well. The novelty of the analysed model is thoroughly discussed in the Introduction to the article and commented upon later in the text.