# Allocation Games

Chapter

## 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 λ_{i}< 0 and*α*_{i}∊ (0,1) such that*α*_{i}(1—exp(-λ_{i}*z*)) 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*= (*x*_{1},...,*x*_{n}) and*y*= (*y*_{1},...,*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=1}^{n}*x*_{i}=*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## Preview

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## Copyright information

© Springer-Verlag Berlin Heidelberg 2000