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## Über dieses Buch

This book is on applications of game theory. The title of this book is not "Game Theory and its Applications" because it does not construct a general theory for considered games. The book contains a lot of examples of applica­ tion of game theory together with the background of those games considered and a list of unsolved problems. Also we consider only the game where the optimal strategies of the players are found in closed form. This book is an attempt to carryon the approach developed in nice books "Search Games" by Gal and "Geometric Games and their Applications" by Ruckle. The first chapter of this book supplies the required definitions and theorems from game theory. The second chapter deals with discrete search games where both players act simultaneously: the games of protection of a channel from infiltration of a submarine, the submarine versus helicopter game, the matrix search games and others. The third chapter considers the game where the players allocate their contin­ uous efforts. In these games players face up an alternative either not to come into contest if the cost of efforts seems too high, or come into it. In the last case the player have to decide how much resources they can afford to spend. The allocation models of search, antiballistic protection and marketing are investigated.

## Inhaltsverzeichnis

### 1. Preliminary Results from Game Theory

Abstract
A two person zero-sum game is defined as a 3-tuple (X, Y, M) where X and Y are sets and M is real valued function defined on the Cartesian product X × Y. The set X is called the set of admissible pure strategies of player 1 and set Y is called the set of admissible pure strategies of player 2. The function M is called the payoff function of player 1. Player 1 chooses a strategy x of the set X while player 2 chooses a strategy y of the set Y. The choices are done simultaneously and independently and the chosen x and y determine the playoff M (x, y) to player 1 and -M(x, y) to player 2. So, it is considered as if each player hands his choice to a referee who then announces (x, y) and executes the payoffs). In zero-sum game the players have antagonistic interests. The payoffs can be considered as amounts of money or utilities. The data of the game (X, Y, M) are known to both players.
Andrey Garnaev

### 2. Ambush Games

Abstract
Consider the following zero-sum infiltration problem. There are two players: Guard and Infiltrator (for example, coast guard and submarine.) Guard wishes to protect a channel against penetration by Infiltrator. For this purpose Guard uses an electric cable with length k. We assume Infiltrator knows the length of cable but it is invisible to him. If Infiltrator crossing the channel touches the cable he gets detected and captured by Guard. Consider the following zero-sum game modelling this problem. Guard chooses in integer interval [1, n] an interval consisting of m points where m < n and Infiltrator chooses an integer point, called his point of crossing, in [1, n]. The payoff to Guard equals 1 if the point chosen by Infiltrator is in the interval chosen by Guard and 0 otherwise. A pure strategy of Guard is denoted by i, where i is his point of crossing and a mixed strategy of Infiltrator is the probability vector x = (x1,..., x n ) saying him to choose his point of crossing at i with probability x i . We denote an integer interval consisting of m points and its left endpoint at p by M (p). Then we can denote a pure strategy y of Guard by p, where p ∊ [1, nm + 1] and p means choosing M(p). A mixed strategy of Guard is the probability vector y = (y1,..., yn-m+1) saying him to choose interval M(i) with probability y i . Let E(x,y) be the expected payoff to Guard if Infiltrator and Guard use the strategy x and y, respectively.
Andrey Garnaev

### 3. Allocation Games

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)
Andrey Garnaev

### 4. Dynamic Infiltration and Inspection Games

Abstract
Consider the following two—person, zero—sum, multi—stage game. There are two players: Guard and Infiltrator. The Infiltrator’s movement is constrained to only integer points: 0,1,2,... on the non—negative x—axis. If Infiltrator is at point i, he may, one unit of time later, move to point i—1, remain at point i or move to point i + 1. Guard, having a gun with k shots, can shoot at most one shot per unit of time at Infiltrator. Infiltrator has a bunker at the origin 0, where he is immune to the Guard’s shooting. It is assumed that Infiltrator knows the number of shots that Guard possesses at all time but Infiltrator does not know where Guard aims his hit and it takes the shot one unit of time to reach x—axis. There is no aiming errors, so Guard can hit any point he desires. If Guard hits the point where Infiltrator locates then the probability of hitting Infiltrator is α where α ∊ (0,1) and 0 otherwise. Therefore, if Guard observes that Infiltrator is at point i, and if Guard desires to shoot at that instant, he should aim at one of the three points i—1, i and i + 1. The payoff to Guard is 1 if he hits Infiltrator and 0 otherwise.
Andrey Garnaev

### 5. Games of Timing

Abstract
5.1 Non-zero Sum Silent Duel
Consider the following non-zero sum duel. Two players (player 1 and 2), starting at time t = 0 at a unit distance before each one’s target, walk toward each one’s target at a constant unit speed with no opportunity to retreat. They will reach their targets at time t = 1. Each player has a gun with one bullet, which may be shot at any time in [0, 1]. Each player selects a time to shoot. The accuracies of shooting are described by the accuracy functions A1(x) and A2(x), where Ai(x) is the probability of hitting by player i his target if he shoots at time x. These functions are differentiable and strictly increasing in [0, 1] such that Ai(0) = 0, Ai(1) = 1 and Ai′(x) > 0 for x ∊ (0, 1), where i = 1, 2. The accuracy functions are fixed and known beforehand to both players. As soon as one of the players hits his target, the contest is over and the first player hitting his target gets payoff 1, and his opponent gets payoff zero. If none of the players hit their targets their payoffs are zero. If both players hit their target at the same time they share their payoffs. Suppose that both players have silent guns, so if player 1 and player 2 shoot at time x and y, respectively, their payoffs are given as follows
$$\begin{array}{*{20}{c}} {{{M}_{1}}(x,y) = \left\{ {\begin{array}{*{20}{c}} {{{A}_{1}}(x)} \hfill & {for x < y,} \hfill \\ {{{P}_{1}}(x)} \hfill & {for x = y,} \hfill \\ {(1 - {{A}_{2}}(y)){{A}_{1}}(x)} \hfill & {for x > y,} \hfill \\ \end{array} } \right.} \\ {{{M}_{2}}(x,y) = \left\{ {\begin{array}{*{20}{c}} {{{A}_{2}}(y)} \hfill & {for y < x,} \hfill \\ {{{P}_{2}}(y)} \hfill & {for y = x,} \hfill \\ {(1 - {{A}_{1}}(x)){{A}_{2}}(y)} \hfill & {for y > x,} \hfill \\ \end{array} } \right.} \\ \end{array}$$
where Pi (x) ∊ [0, Ai(x)) for x ∊ (0, 1], Pi(0) = 0, for example, if the players share the payoff equally when they both hit the targets, then
Andrey Garnaev

### 6. Parlour Games

Abstract
Baston and Bostock [11] considered the following zero-sum game. There are two players, say 1 and 2. Two numbers are chosen from [0, 1] independently by means of uniform distributions. Player 1 looks at the numbers privately, chooses one of the two and opens it to his opponent. Player 2 then accepts either the opened number or the covered (unopened) one, and receives from player 1 his accepted number. Baston and Bostock showed that the optimal strategy of player 1 is to choose the nearest to 1/2 number and open it, the optimal strategy of player 2 is to accept the opened number if and only if it is at least 1/2. The value of the game is 7/12 which is greater than 1/2.
Andrey Garnaev

### Backmatter

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