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A [[Game involving the choice of the moment of time|game involving the choice of the moment of time]] describing the following type of conflict. Two opponents may shoot at each other during a certain period of time, while the weapons at their disposal have only a limited number of rounds of ammunition. The strategies of the players are the moments of time chosen for firing. The pay-off function is defined as the mathematical expectation of a certain random variable which assumes a finite number of values, corresponding to the results of the duel. Depending on the information about the activities of the opponent, duels may be noisy or noise-less (silent). For instance, if each player can fire one shot, and if that shot must be fired in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034140/d0341401.png" />, if the accuracy functions (i.e. the hit probabilities of the players I and II) are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034140/d0341402.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034140/d0341403.png" />, respectively; then, if player I gains 1 point if he kills II, loses 1 point if he is killed himself, and the gain is 0 in all other cases, the pay-off functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034140/d0341404.png" /> are:
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A [[Game involving the choice of the moment of time|game involving the choice of the moment of time]] describing the following type of conflict. Two opponents may shoot at each other during a certain period of time, while the weapons at their disposal have only a limited number of rounds of ammunition. The strategies of the players are the moments of time chosen for firing. The pay-off function is defined as the mathematical expectation of a certain random variable which assumes a finite number of values, corresponding to the results of the duel. Depending on the information about the activities of the opponent, duels may be noisy or noise-less (silent). For instance, if each player can fire one shot, and if that shot must be fired in the interval $[0,1]$, if the accuracy functions (i.e. the hit probabilities of the players I and II) are $p(x)$ and $q(y)$, respectively; then, if player I gains 1 point if he kills II, loses 1 point if he is killed himself, and the gain is 0 in all other cases, the pay-off functions $K(x,y)$ are:
  
 
in a noisy duel
 
in a noisy duel
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034140/d0341405.png" /></td> </tr></table>
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$$K(x,y)=\begin{cases}p(x)-(1-p(x))\sup_{y>x}q(y),&x<y,\\p(x)-q(x),&x=y,\\-q(y)+(1-q(y))\sup_{x>y}p(x),&x>y;\end{cases}$$
  
 
in a silent duel
 
in a silent duel
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034140/d0341406.png" /></td> </tr></table>
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$$K(x,y)=\begin{cases}p(x)-(1-p(x))q(y),&x<y,\\p(x)-q(x),&x=y,\\-q(y)+(1-q(y))p(x),&x>y.\end{cases}$$
  
 
The duels under study include duels in which the opponents have several shots at their disposal or are permitted to expend the available resources in a continuous manner.
 
The duels under study include duels in which the opponents have several shots at their disposal or are permitted to expend the available resources in a continuous manner.

Revision as of 10:55, 9 November 2014

A game involving the choice of the moment of time describing the following type of conflict. Two opponents may shoot at each other during a certain period of time, while the weapons at their disposal have only a limited number of rounds of ammunition. The strategies of the players are the moments of time chosen for firing. The pay-off function is defined as the mathematical expectation of a certain random variable which assumes a finite number of values, corresponding to the results of the duel. Depending on the information about the activities of the opponent, duels may be noisy or noise-less (silent). For instance, if each player can fire one shot, and if that shot must be fired in the interval $[0,1]$, if the accuracy functions (i.e. the hit probabilities of the players I and II) are $p(x)$ and $q(y)$, respectively; then, if player I gains 1 point if he kills II, loses 1 point if he is killed himself, and the gain is 0 in all other cases, the pay-off functions $K(x,y)$ are:

in a noisy duel

$$K(x,y)=\begin{cases}p(x)-(1-p(x))\sup_{y>x}q(y),&x<y,\\p(x)-q(x),&x=y,\\-q(y)+(1-q(y))\sup_{x>y}p(x),&x>y;\end{cases}$$

in a silent duel

$$K(x,y)=\begin{cases}p(x)-(1-p(x))q(y),&x<y,\\p(x)-q(x),&x=y,\\-q(y)+(1-q(y))p(x),&x>y.\end{cases}$$

The duels under study include duels in which the opponents have several shots at their disposal or are permitted to expend the available resources in a continuous manner.

References

[1] S. Karlin, "Mathematical methods in the theory of games, programming and economics" , Addison-Wesley (1959)
How to Cite This Entry:
Duel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Duel&oldid=14860
This article was adapted from an original article by E.B. Yanovskaya (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article