# Allocation Games

• Andrey Garnaev
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 485)

## Abstract

3.1 One-Sided Allocation Game Without Search Cost

Consider 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 λi < 0 and αi ∊ (0,1) such that αi(1—exp(-λiz)) 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,..., xn) and y = (y1,..., yn), respectively, where yi is the probability that Hider hides in box i and xi is the amount of effort allocated in box i by Searcher, where xi ≥ 0 for i ∊ [1, n] and ∑ i=1 n xi = X. So, the payoff to Searcher if Searcher and Hider employ strategies x and y, respectively, is given by
$$M(x,y) = \sum\limits_{i = 1}^n {\alpha _i y_i } \left( {1 - exp\left( { - \lambda _i x_i } \right)} \right).$$
(1)

### Keywords

Marketing Expense Nash Librium Alloca

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