SummaryIn this first Learning Path on Game theory, we learn about the main tools and conditions required in order to make a thorough analysis of games. We see how the quality of information shape the way we solve games, and learn about how to describe them.
In game theory, the extensive form is away of describing a game using a game tree. It’s simply a diagram that shows that choices are made at different points in time (corresponding to each node). The payoffs are represented at the end of each branch. Since the extensive form represents decisions at different moments, it’s usually used to describe sequential games, while simultaneous games are described using the strategic form. Since sequential games imply making decisions at different moments for each player, information is perfect since each player can see the decision taken by the previous player, complete and the rules of the game and each player’s payoffs are common knowledge.
In the first game tree we can see how player 1 is the first to decide, while player 2 will make a decision after observing what player 1 has decided. The payoffs represented at the end of each brand represent all possible outcomes. For instance, if player 1 chooses strategy A and player 2 chooses strategy B, the set of payoffs will be p1A,p2B.
A good example of a sequential game described with the extensive form is when considering collusion agreements, as depicted in the second game tree.
Two firms share the market, colluding and maintaining high prices. Each firm can decide to stop colluding and start a price war, in order to increase their market share, even force the other to quit the market. Firm 1 can either keep colluding with firm 2, or start a price war. If firm 1 decides to keep colluding, firm 2 will need to make a decision. If they both agree to collude, they will get 5,5. However, if one of them decides to start a price war, the set of payoffs will be either 4,3 or 3,4, depending on which one starts the war (and therefore acquires a greater market share). It’s easy to see that collude-collude is both the Nash equilibrium and a Pareto optimum situation. This result may change when considering repeated games.
It’s worth mentioning that the extensive form can be used also to describe simultaneous games, by using information sets, as shown in the third game tree. These information sets, usually represented by a dashed line uniting two nodes or by encircling them, mean that the player does not know in which node he is, which implies imperfect information, like when using the strategic form.