Discrete Games Research Articles

Differential Offensive-Defensive Games between Plants and Herbivores

Abstract A mathematical model of chemically mediated plant/herbivore interaction dynamics is developed in the context of the assumptions basic to Optimal Foraging and Optimal Defense Theories (Rhoades). Evidently, plants produce some important defensive chemicals in allometric proportion to biomass, and are also capable of spontaneous response to damage caused by physical stress and/or herbivore attack, by actually increasing percentages of these chemicals in plant tissues over as short a time as a few hours. In addition, herbivores have evolved offensive strategies to counter the plants' defenses. Our mathematical theory centers on a differential zero-sum game in production /consumption space. However, the associated discrete game with noise and particular sequences of play , is also considered. The concept of a zero-sum production/consumption game as a representation of an interactive coevolved two species predator-prey system is basic to our approach. The dynamics of the game are developed in a series of biological arguments ending finally with a pair of second order, nonlinear, coupled, symmetric Duffing oscillators with variable second order damping. Plant vigour or resistance may be studied in terms of properties of this damping which depends upon chemical response levels in plant tissues. Both plant and herbivore players use bang-bang controls and automatic feedback programs. To be explicit, the plant controls are the Rhoades' Response Parameter (ν) and the coefficients of palatability (u ⌝δ ). The herbivore's controls are its coefficient of aggregation (γ) and its coefficient of foraging efficiency (u 2 δ). The parameter δ has been proved identifiable by us, elsewhere. As an outcome of this theory locusts and larch budmoths, can be successfully modelled as alternating herbivore strategies, first stealthy γ = 0 and then Opportunistic γ ⪢ 0, in a single zero-sum (production/consumption)-game . Thus, the (Larch/Larch budmoth)-system is quite different from the “cooperative behaviour” of limit cycle models, when viewed from the zero-sum game perspective.

The sequence method for finding solutions to infinite games: A first demonstrating example

A noncooperative infinite game can be approached by a sequence of discrete games. For each game in the sequence, a Nash solution can be found as well as their limit. This idea and procedure was used before as a theoretical device to prove existence of solutions to games with continuous payoffs and recently even for a class of games with discontinuous ones (Dasgupta and Maskin, 1981). No one, however, used the method for the actual solution of a game. Here, an example demonstrates the method's usefulness in finding a solution to a two-person game on the unit square with discontinuous payoff functions.

Irregular gaming areas

This work deals with those discrete games, the corresponding payoff matrices of which might be changed according to some unknown law within a given period of time. Such games are often seen in irregular gaming areas where some existing situations can act upon the elements of the payoff matrix. A computational algorithm is proposed, and a new management game under catastrophe conditions pertaining to payoff matrix is given. Graph theory, theory of Games, simulation and Boolean algebra are used.

Player information in discrete games

Player information in discrete games

Nonlinear discrete escape games

Nonlinear discrete escape games

Nash Equilibrium Strategies for the Problem of Armament Race and Control

The problem of modeling and “optimally controlling” an arms race situation between two nations is a very interesting application, of differential game theory. In this pacer, armament, buildup between two nations is modeled as a multistage game problem. The closed-loop Nash, strategy is suggested as a possible solution concept for this problem and it is shown that if the cost functions are quadratic, then the system together with its closed-loop Nash strategies will have the same structure as the well-known Richardson's model of arms race. Consequently, it is suggested that the coefficients in Richardson's model can be determined (or predicted) based on the solution of this multistage discrete game problem. A computational procedure based on the solution of a set of recursive equations is also presented. This type of analysis is not only useful for guiding the armament decision-making process of a nation in an arms race with another nation but also is helpful in assessing economic and military aids by a large nation to smaller nations in an arms race.

On a certain discrete pursuit game on graphs

On a certain discrete pursuit game on graphs

Additional aspects of the Stackelberg strategy in nonzero-sum games

The Stackelberg strategy in nonzero-sum games is a reasonable solution concept for games where, either due to lack of information on the part of one player about the performance function of the other, or due to different speeds in computing the strategies, or due to differences in size or strength, one player dominates the entire game by imposing a solution which is favorable to himself. This paper discusses some properties of this solution concept when the players use controls that are functions of the state variables of the game in addition to time. The difficulties in determining such controls are also pointed out. A simple two-stage finite state discrete game is used to illustrate these properties.

A class of linear discrete qualitative games

A class of linear discrete qualitative games

Noncooperative N-person game theory (N≥2) using expected relative desirability criterion

Considered is noncooperative discrete N-person game theory forN≥2 (where the players choose their strategies separately and independently). Payoffs can be of a very general nature and are not necessarily numbers. However, each player is able to quantitatively specify the relative desirability of the possible game outcomes (N-dimensional, a pay off to each player) according to his preferences. That is, he specifies a positive number for each outcome and these numbers are such that their ratios quantitatively represent the relatively desirability of the corresponding outcomes to this player. For each player, the criterion is the expected relative desirability to him of what occurs for the game, and an optimum mixed strategy maximizes the minimum value of this criterion over all possible mixed strategies that could be used by the other players. An optimum solution is obtained by used of classical minimax game theory. Practical implications of applying thisN-person game theory are examined. Also, a possible approach for developing quantitative functions to provide the relative desirability numbers is outlined.

Linear discrete qualitative games

Linear discrete qualitative games

Some aspects of nonzero-sum differential games

Some properties of nonzero-sum differential games are explored. A differential game is defined in extensive form, and the set of attainable payoffs P is defined. Various subsets of P which are useful in discussing solutions are identified. The concept of signaling in discrete games is interpreted as dynamic bargaining in continuously evolving games. Via these concepts a player can, at little expense to himself, attempt to induce the other players to adopt a strategy more favorable to himself. Several examples are given to illustrate the definitions and developments. Interpreting some classical matrix games as continuously evolving games helps to clarify that there can be cooperation or threatening without pregame or explicit negotiation and bargaining. A simple two-player differential game which has been programmed on a hybrid computer and for which some limited experimental results are presented is also described.

Further properties of nonzero-sum differential games

The general nonzero-sum differential game hasN players, each controlling a different set of inputs to a single nonlinear dynamic system and each trying to minimize a different performance criterion. These general games have several interesting features which are absent in the two bestknown special cases (the optimal control problem and the two-person, zero-sum differential game). This paper considers some of the difficulties which arise in attempting to generalize ideas which are well known in optimal control theory, such as theprinciple of optimality and the relation betweenopen-loop andclosed-loop controls. Two types of solutions are discussed: theNash equilibrium and thenoninferior set. Some simple multistage discrete games are used to illustrate phenomena which also arise in the continuous formulation.