Part VI - Dynamic games

Summary

INTRODUCTION

In Chapter 18, we introduce extensive form games, which describe the potential evolution of a dynamic game using a game tree. In each node of the tree some player (or players) should choose their action(s), and each branch emanating from such a node stands for an action (or action profile) that the player(s) active at that node can choose. Each leaf of the tree – a node from which no further branches extend – stands for a possible termination point of the dynamic interaction, and specifies the payoffs of the players upon this termination. Each leaf corresponds uniquely to a path of consecutive branches from the root to that leaf.

A strategy of a player in an extensive form game is a complete plan of action, which specifies what the player would do at each of the nodes where she may be called upon to play, if and when that node is ever reached. Each profile of the players’ strategies defines a path from the root to a leaf specifying the payoffs of the players. Thus, associating each strategy profile of the players with these payoffs defines the strategic form of the extensive form game. The Nash equilibria of this strategic form are the Nash equilibria of the extensive form game.

The strategic form abstracts from the dynamic aspect of the game. As a result, some extensive form games have implausible Nash equilibria, based on a non-credible threat – an action that the player wouldn’t actually like to take upon reaching his time to decide, but that at the Nash equilibrium the player doesn’t need to take anyway, because due to the fear from this incredible threat a player acting prior to him preempts altogether by deviating the game to another path; this alternative path may be beneficial for the player who posed the non-credible threat but detrimental to the preempting player who acted before him.