Allocation Games

3.1 One-Sided Allocation Game Without Search CostConsider the following zero-sum one-sided allocation game on integer interval [1, n]. Hider selects one of the n points and hides there. Searcher seeks Hider by dividing the given total continuous search effort X and allocating it in each point. Each point i is characterized by two detection parameters λ_i < 0 and α_i ∊ (0,1) such that α_i(1—exp(-λ_iz)) is the probability that a search of point i by Searcher with an amount of search effort z will discover Hider if he is there. The payoff to Searcher is 1 if Hider is detected and 0 otherwise. A strategy of Searcher and Hider can be represented by x = (x₁,..., x_n) and y = (y₁,..., y_n), respectively, where y_i is the probability that Hider hides in box i and x_i is the amount of effort allocated in box i by Searcher, where x_i ≥ 0 for i ∊ [1, n] and ∑_i=1nx_i = X. So, the payoff to Searcher if Searcher and Hider employ strategies x and y, respectively, is given by 1$$ M(x,y) = \sum\limits_{i = 1}^n {\alpha _i y_i } \left( {1 - exp\left( { - \lambda _i x_i } \right)} \right). $$