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Int J Game Theory (2006) 34:185205
DOI 10.1007/s00182-006-0013-xORIGINAL ARTICLEThe existence and uniqueness of Nash equilibrium
point in an m-player gameShoot later, shoot first!E. Presman I. SoninReceived: 26 July 2002 / Revised: 26 September 2004,
1 October 2005 / Accepted: 25 January 2006 /
Published online: 2 June 2006 Springer-Verlag 2006Abstract We consider the following silent duel of m players with a possible
economic interpretation. Each player has one bullet, which she can shoot at
any time during the time interval [0, 1]. The probability that the i-th player hits
the target at moment t is given by an increasing accuracy function fi(t).The
winner is the player who hits the target first. Under natural assumptions on the
functions fi(t) we prove the existence and uniqueness of a Nash equilibrium
point in this game, and we provide an explicit construction of this equilibrium.
This construction allows us to obtain exact solutions for many specific examples.
Some of them are presented.Keywords Duel of m players Nash equilibrium1 IntroductionWe consider the game of m players with a possibly broad interpretation, but
first we formulate it as follows. Each of m players is initially located at a distance from a target. It can be a common target or each player has her/his own
target, or in the case of two players, each serves as a target for the other. At
the moment t = 0 each player starts to move towards the target. Each player This work was partly supported by RBRF grants 03-01-00479.E. Presman (B)CEMI, Russian Academy of Science, Nakhimovskii Pr. 47,
Moscow 117418, Russiae-mail: [email protected]. SoninDepartment of Mathematics, UNC Charlotte, Charlotte,
NC 28223, USAe-mail: [email protected] E. Presman, I. Soninhas one bullet, which she/he can shoot at any time during the time interval
[0, 1]. The probability that the ith player hits the target at moment t is given
by an accuracy function fi(t), where fi(t) is a continuous, (strictly) increasing
function, 0 fi(t) 1, and fi(0) = 0. The winner of the game is the player who
hits the target first. If two or more players hit the target at the same time, the
winner is determined by a lottery (draw rule) among these players. The rules
of the lottery might depend on time, the probability that nobody...