We are going to switch gears and analyze sequential games. The format will be different but the principles of game theory that we reviewed will be the same. I am going to introduce the concept of backward induction to solve these type of games. Similar to simultaneous games, the sequential games will be complete and with perfect information.
Let’s look at the following example in extensive form:
Going forward I will illustrate extensive form games in a game tree as shown above. In this example we have two siblings, one age 4 and the other age 2. On the table there is a plate with only two cookies remaining. Player 1 represents the older sibling and chooses an action a¹ from his or her set of options A¹. In this game the older sibling has the actions of “share” or “take”. Player 2 represents the younger sibling and observes the action of the older brother and then chooses an action a² (“share” or “take”) from his or her set of options A².
So if the older sibling chooses “take” then the payoff will be two cookies for the older sibling and none for the younger sibling (how cruel). If the older sibling chooses the option of “share”, then the younger sibling can choose either “take” or “share.” Now we can introduce the concept of backwards induction. We basically look at the options of the younger sibling first and work our way backwards.
Before we start, it is important to remember that players will look for the optimal choice based on their sets of options (similar to the simultaneous games we analyzed). So the younger sibling has the option of choosing the action “share” or “take”. If the younger sibling chooses “share” then both siblings will end up with one cookie. If the younger sibling chooses “take” then the younger sibling gets two cookies and the older sibling gets nothing. Since we are assuming rational players, the younger sibling will choose “take” since 2 > 1.
Knowing that the younger sibling will choose “take” the optimal choice for the older sibling is to choose “take” since 2 > 0. So this game ends where the older sibling will approach the table and grab both cookies since it is assumed the younger sibling would do the same thing given the opportunity. It appears that both siblings do not understand the concept of sharing and equality. Although a simple example, this game represents the basics of backward induction and solving sequential games with complete information.
Going forward we will look at larger games with more players, actions, and options. I look forward to your feedback in the comments below and feel free to follow me on Twitter and connect via LinkedIn or Facebook or both.