# 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

Nash Equilibrium Optimal Strategy Search Effort Market Game Unique Nash Equilibrium
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.