4 Presentations

• 
  10:30 AM - 11:00 AM

  The Return Function: A new Tool to Compute Bayesian-Nash Equilibria

  • Lê Nguyên Hoang, presenter, GERAD - Polytechnique Montréal
  • Francois Soumis, GERAD et Polytechnique
  • Georges Zaccour, Chair in Game Theory and Management, GERAD and HEC Montréal

  In this presentation, we introduce a new mathematical tool to compute and study Bayesian-Nash equilibria, called the return function. This function maps players' strategies with the induced probabilistic distribution of the outcomes. The return function depends on the setting of the game, the probabilistic distribution of the types of players and on their strategies. Given a return function, players can evaluate their best actions, which corresponds to choosing the best reply strategy to other players' strategies. By simulating a best-reply iteration process similar to fictitious play (but using best replies to the return function rather than best replies to others' previous actions), we propose an algorithm which computes Bayesian-Nash equilibria. In particular, we discuss properties and convergence of return functions in this process.

• 
  11:00 AM - 11:30 AM

  The Evolution of Cooperation with Action-Varying Functions for Strategy Copying Using Cellular Automata

  • Pedro Henrique Triguis Schimit, presenter,
  • Buruthagas Santos, Universidade Nove de Julho
  • Carlson Soares, Universidade Nove de Julho

  In this work, we use cellular automata to model a population which the cells represent individuals that interact with their neighbors playing a game. The lattice has N=100, NxN = n² individuals which interacts with the 8 surrounded individuals in the form of a Moore neighborhood [1]. Those games between two individuals may be either considered as a competition for a benefit or resource.

  The individuals begin the simulation with an amount of life of 100. When a game is played, the payoff of each individual is calculated according to its action and the amount of life decreases. When the amount of life is equal or lower than zero the individual dies and a new one replaces, maintaining the NxN = n² individuals in the population constant. The advantage of such approach is that each player plays with 8 different individuals separately, not as a multi-player matrix game [2].
  The games played have the form of Prisoner’s Dilemma and Hawk-Dove game [3]. For the first game, the individuals can cooperate or defect. The probability of cooperation of each individual is assigned when he/she born and it is considered as him/her strategy. For the Hawk-Dove game, the individuals can either act like a hawk, attacking the opponent, or dove that runs away of the fight. The probability of acting like a dove of each individual is assigned when he/she born and it is considered as him/her strategy [4].

  The simulations begin with the probability of each player cooperate (or act like a dove) randomly chosen from 0 to 1. When the individual dies, the new individual that will replace the dead one must copy a strategy from their neighbors. Here is the question: How to evaluate the better strategy? The model suggests three ways for the new individual to choose: 1- He/She copies the strategy from the individual with the highest amount of life; 2 – He/She copies the strategy from the individual with the highest life, that is, the older individual in the neighborhood; 3 – He/She copies the strategy from the individual with highest R, where R is defined as follow: R=(lifetime*amount of life)/(caused death). The caused death is the amount of life that individual took off the players which he/she played during life.
  The results show that cooperation arises in the cases 1 and 2, that is, the most part of population have a strategy more likely to cooperate (or act like a dove). The case 3 have the highest cooperation level.

  The work tries to show that the evaluation of strategies by individuals to copy alters the macro behavior of population. The cooperation level in population may have different levels according to the function chosen to copy a strategy [5].

  Keywords: cellular automata, evolution of cooperation, game theory, hawk-dove, prisoner’s dilemma.

  Bbiliography

  [1] Wolfram, S., 1994. Cellular Automata and Complexity: Collected Papers. Westview Press, New York.
  [2] Broom, M., Cannings, C., Vickers, G.T., 1997. Multi-player matrix games. Bull. Math. Biol. 59 931-952.
  [3] Smith, J.M., 1982. Evolution and the Teory of Games. Cambridge University Press.
  [4] Szabó, G., Fáth, G., 2007. Evolutionary games on graphs Physics Reports 446 97 – 216.
  [5] Lieberman, E., Hauert, C. & Nowak, M.A., 2005. Evolutionary dynamics on graphs. Nature 433 312–316.

• 
  11:30 AM - 12:00 PM

  Coalition Strategies in Syndicated Loans

  • Pascal Francois, presenter, HEC Montreal
  • Michèle Breton, GERAD, HEC Montréal

  The syndicated loan market is ruled by decisional interactions among different actors. In this paper, we analyze the behavior of banks and coalition formation to structure syndicated loans from a strategic standpoint. In a game theory framework, we use a sequential coalition formation process to model the way banks enter into an agreement to participate in a syndicated loan. Using reputation mechanisms, the proposed dynamic programming approach, coupled with maximization of utility functions, aims at leading to efficient coalitions, in the sense of the role and the loan share assigned to the coalition members as well as the respective decided level of effort. Computational results presenting the performance of the algorithm are reported and a sensitivity analysis is performed on the most influential model parameters.

• 
  12:00 PM - 12:30 PM

  Evolutionary Games and Local Dynamics

  • Philippe Uyttendaele, presenter, Maastricht University
  • Frank Thuijsman, Maastricht University

  In the classical approach of evolutionary game theory, the so called replicator dynamics are the driving force behind the development of a population in time. In such models the population usually consists of a continuum of individuals that each belong to a finite number of types. Individuals meet randomly and these interactions have an impact on the fitness of their types. The fitness payoffs are given in a fitness matrix. Types that are doing better than average increase in number, while the types that are doing worse decrease. The fact that individuals meet randomly implies that the population is homogeneously distributed and the location of these individuals plays no role. In this paper we examine what happens when individuals do have a specific location and only interact with their local neighborhood.
  With the help of a few examples, we show that there is a major difference when using local interaction instead of a global dynamical system. For example the symbiosis of two types to sustain in a hostile environment is now possible. Also we show that the choice of the local interaction schema is very wide and can drive to many different kind of results. With the help of Cellular Automata, we want to bridge the field of dynamical systems and biology.