TIME-UNIT SHIFTING IN 3-PERSON GAMES IN FINITE AND UNCOUNTABLY INFINITE STAIRCASE-FUNCTION SPACES SOLVED IN PURE STRATEGIES
DOI:
https://doi.org/10.20535/kpisn.2025.1.321883Keywords:
game theory; payoff functional; 3-person game; staircase-function strategy; trimatrix game; staircase-function equilibrium pure strategy.Abstract
Background. Games played with staircase-function pure strategies can model discrete-time dynamics of rationalizing the distribution of some limited resources among players. Along with 2-person games, 3-person games are the most applicable models of rationalization in economics, ecology, social sciences, politics, government, and sports. There is a known method of finding an equilibrium in a 3-person game played in staircase-function pure strategy spaces. The time interval on which the game is defined consists of an integer number of time units. The equilibrium is stacked from time-unit equilibria. An open problem is a multiplicity of equilibria (on some time units) leading to a multiplicity of equilibrium stacks. Another open question is how to deal with a 3-person game in which the time interval can be changed or shifted by an integer number of time units.
Objective. The purpose of the paper is to expand and develop the tractable method of solving 3-person games played within players’ finite sets of staircase functions for the case when the length of the time interval on which the 3-person game is defined is varied by an integer number of time units.
Methods. To achieve the said objective, a 3-person game, in which the players’ strategies are staircase functions of time, is formalized. In such a game, the set of the player’s pure strategies is a continuum of staircase functions. The time can be thought of as it is discrete due to the time interval is comprised of time units (subintervals). Then the set of possible values of the player’s pure strategy is discretized so that the player possesses a finite set of staircase functions.
Results. The known method is expanded to build a single pure-strategy equilibrium stack in a discrete-time staircase-function 3-person game. The criterion for selecting a single equilibrium solution is to maximize the players’ payoff sum. In the case of a time-unit shifting, this criterion allows extracting the respective best staircase-function equilibrium pure strategy of the player in any “narrower” subgame from the player’s best staircase-function equilibrium pure strategy in the “wider” game.
Conclusions. A tractable and efficient method of finding the best pure-strategy equilibrium in a 3-person game played in finite or uncountably infinite staircase-function spaces is to solve a succession of time-unit 3-person games, whereupon their best equilibria are stacked into the best pure-strategy equilibrium. To deal with the case when not every time-unit 3-person game is solved in pure strategies, an effective way is to put a staircase-function game on hold-up on those time units which do not have pure-strategy equilibria. The result of putting the staircase-function game on hold-ups is that the player will obtain one’s best staircase-function equilibrium pure strategy with gaps, whichever the time interval and time-unit shifting are.
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