Discrete Strategy Sets Research Articles

Half a century of evolutionary games: a synthesis of theory, application and future directions.

Half a century of evolutionary games: a synthesis of theory, application and future directions.

A mathematical interpretation of the communicative constitution of organizations

ABSTRACT We propose a probabilistic control-system model to measure gradations of organizational coherence. Our model expresses the four flows of the Communicative Constitution of Organizations, and moves beyond common network-based analyses to quantify the emergence of organization through communication. In the model, an organization’s leadership selects from a set of discrete strategies, the number and complexity of which are regulated by membership negotiation and self-structuring. Activity coordination constrains the internal response of the organization to the selected strategy, and institutional positioning shapes its ultimate outcome in the environment. Our model introduces a new partition of the four flows between those that affect the set of available strategies versus the precision of organizational control. The model informs proxy measurements of the maturation of a partial organization, such as an insurgency. Its quantified approach enables testing of the interrelationships and tradeoffs among flows in an agent-based simulation of varying levels of partial organizations competing and cooperatively exploring for resources. The most consequential flow in our scenario, measured in comparison to baseline data, is membership negotiation. Complex behaviours emerge in simulation results, such as information asymmetries, tradeoffs between power and efficiency, the contrast between short- and long-term gains, and individual versus group utility.

Analysis and Interventions in Large Network Games

We review classic results and recent progress on equilibrium analysis, dynamics, and optimal interventions in network games with both continuous and discrete strategy sets. We study strategic interactions in deterministic networks as well as networks generated from a stochastic network formation model. For the former case, we review a unifying framework for analysis based on the theory of variational inequalities. For the latter case, we highlight how knowledge of the stochastic network formation model can be used by a central planner to design interventions for large networks in a computationally efficient manner when exact network data are not available.

Analysis and Interventions in Large Network Games

We review classic results and recent progress on equilibrium analysis, dynamics, and optimal interventions in network games with both continuous and discrete strategy sets. We study strategic interactions in deterministic networks as well as networks generated from a stochastic network formation model. For the former case, we review a unifying framework for analysis based on the theory of variational inequalities. For the latter case, we highlight how knowledge of the stochastic network formation model can be used by a central planner to design interventions for large networks in a computationally efficient manner when exact network data are not available.

All-pay auctions with asymmetric effort constraints

We study all-pay auctions with discrete strategy sets and analyze the equilibrium strategies when players have asymmetric values of winning as well as asymmetric effort constraints. We prove that for any number of players if one of them has the highest effort constraint then, independent of the players’ values of winning, he is the only player with a positive expected payoff. However, when two players have the same highest effort constraint then they do not necessarily have the highest expected payoffs. By several examples we show a significant distinction between the equilibrium strategies of two players and a larger number of players, particularly when the player with the highest effort constraint is not unique.

Learning algorithms for joint resource block and power allocation in underlay D2D networks

We investigate the problem of resource block (RB) and power allocation jointly and in a distributed manner using game theoretic learning solutions, in an underlay device-to-device network where device pairs communicate directly with each other by reusing the spectrum allocated to the cellular users. We formulate the joint RB and power allocation as multi-agent learning problems with discrete strategy sets; and suggest partially distributed and fully distributed learning algorithms to determine the RB and power level to be used by each device pair. The partially distributed algorithms, viz., Fictitious Play and its variant Fading Memory Joint Strategy Fictitious Play with Inertia, achieve Nash Equilibrium (NE) of the sum-rate maximization game in a static wireless environment. The completely distributed and uncoupled Stochastic Learning Algorithm converges to pure strategy NE of the interference mitigation game in a time-varying radio environment. We provide proofs for the existence of NE and convergence of the learning algorithms to the NE. Performance of the proposed schemes are evaluated in log-normal, Rayleigh and Nakagami fading environments and compared with an existing hybrid scheme and a centralized scheme. The simulation results show that the partially distributed schemes give the same performance as the centralized scheme, and the fully distributed scheme gives similar performance as the hybrid scheme but with much reduced signaling and computation overhead.

Co-evolutionary approach for strategic bidding in competitive electricity markets

Determining optimal bidding strategies in a competitive electricity market to maximize the profit of each bidder is a challenging economic game problem. In this paper, it is formulated as a bi-level optimization problem in which, in the lower level, the community’s social welfare is maximized by solving a power flow problem while, in the upper level, the profits of individual bidders are maximized. In this bidders’ game, instead of using a set of discrete strategies as is usual, we consider continuous functions as strategies. To solve the upper-level problem, two co-evolutionary approaches are proposed and, for the lower level, an interior point algorithm is applied. Three IEEE benchmark problems in four different scenarios are solved and their results compared with those obtained from two conventional approaches and the literature which indicate that the proposed approaches have some merit regarding quality and efficiency.

Computing All Solutions of Nash Equilibrium Problems with Discrete Strategy Sets

The Nash equilibrium problem is a widely used tool for modeling noncooperative games. Many solution methods have been proposed in the literature to compute solutions of Nash equilibrium problems with continuous strategy sets, but, aside from some specific methods for some particular applications, there are no general algorithms to compute solutions of Nash equilibrium problems in which the strategy set of each player is assumed to be discrete. We define a branching method to compute the whole solution set of Nash equilibrium problems with discrete strategy sets. This method is equipped with a procedure that, by fixing variables, effectively prunes the branches of the search tree. Furthermore, we propose a preliminary procedure that by shrinking the feasible set improves the performance of the branching method when tackling a particular class of problems. Moreover, we prove existence of equilibria and we propose an extremely fast Jacobi-type method which leads to one equilibrium for a new class of Nash equ...

DISCRETE CONCAVITY FOR POTENTIAL GAMES

This paper proposes a discrete analogue of concavity appropriate for potential games with discrete strategy sets. It guarantees that every Nash equilibrium maximizes a potential function.

A discrete fixed point theorem and its applications

This paper proves a fixed point theorem on a discrete set which exploits the “contiguous convexity” of the set and the “direction preserving-ness” of correspondence. Then, applying this result, we study a general equilibrium model with indivisible commodities. A set of sufficient conditions for the existence of a Walrasian equilibrium price is given, in terms of properties of the excess demand function. We also apply it to a non-cooperative game on discrete strategy sets, and show some conditions for the existence of a Nash equilibrium point. A new notion of “contiguity” is proposed as a discrete analog of continuity.