Most of the games I have described so far could be categorized as games that are played simultaneously. Regardless of whether it is rock-paper-scissors game or the prisoner’s dilemma, each player can anticipate his/her opponent’s response to some extent, but does not know his/her strategy for sure. Even though the notion of Nash equilibrium provides a reasonable way for people to surmise each other’s strategy under the assumption that everyone is rational, each player still cannot conclusively determine what strategies his/her opponent will play until his/her opponent has played it. Now, the course shifts its focus into sequential games, where a player can make a decision based on the strategy that his/her opponent has played already.
Since this is a completely new idea, I will use an example to elucidate it. I will focus more on explaining the theory in this blog and elaborate more on the purpose of this reasoning in future blogs.
Suppose I am the commander of an army and want to invade an island that Jerry is ruling. When my army arrives at the island, Jerry has two options. He can choose either to fight or to run away. Regardless of the strategy he chooses, I have two options as well: to fight or to run away. Instead of constructing a payoff matrix like we did before, we can make a tree graph in order to see the situation better.
As you can see in my tree graph, if we both choose to fight with each other, neither of us is better off, thus respectively receiving a payoff of 0 and 0. If Jerry chooses to fight, and I choose to run away, he will receive a payoff of 2, and I will receive a payoff of 1, because at least, some of my army will survive in the battle. However, if Jerry chooses to run away, and I choose to fight, I will receive a payoff of 2, and he will receive a payoff of 1, because I have the advantage of fighting an opponent who is running away. If I choose to run away, I will receive a payoff of 1, because most of my army will remain, and Jerry will receive a payoff of 2, because even though Jerry runs away, he successfully protects his territory.
By using this tree graph, we can use backward induction method, which is a way to analyze a situation by thinking backwards and to find out what the final outcome is. If Jerry fights, I will choose to run away, but if Jerry chooses to run away, I will choose to fight. However, this reasoning is rather clear to Jerry. Jerry will think that if he fights, Joe will choose to run away, but if he runs away, Joe will choose to fight. Therefore, Jerry will choose to fight by thinking this situation backward.
This situation put me in trouble, because Jerry gets a higher payoff than I do. What can I do to overturn this situation?
I can make a commitment to burning my ships so that there is no way for my soldiers to run away!
Then my tree graph will become like this:
I successfully delete a strategy that might eventually put me in disadvantage. But now, I have to fight no matter what Jerry chooses. The eventual equilibrium outcome will yield a payoff of 0 and 0, which is not ideal for me, but it is even less ideal for Jerry.
The lesson we can learn here is that I can be better off by getting rid of choices! This game is called a commitment game: people are better off by making commitments, because that narrows down available choices to choose from. The time wasted on determining whether I should put effort into learning English can be used more effectively used if I am committed to learning English starting from now. Then, I do not need to consider what else I should do other than learning English, and thus makw progress on my English ability. This is true in today’s society nowadays, for there are more and more choices. If you are ever bogged down by a myriad of choices, think about other potential choices and try to make commitment to one the choices. It will make your life easier, and potentially, better.
This game theory example elucidates the real life importance of making commitment to narrow down choices, for it can bring good outcome.
An article I read recently on the Prisoner’s Dilemma takes into the account of sequential moves.