Note that for purposes of this node I will assume you already know the rudiments of game theory. Check that node out before reading this.
Game theory has two standard models for explaining human interaction. The normal form game, also called the simultaneous game, assumes that both players make their moves at the same time. This is the most commonly used model, and is the type used as an example in the Game theory node.
The sequential game (or extended-form game) assumes exactly the opposite: One player moves first. Only then, after having seen the first player's move, does the second decide on an action. This allows for player 2 to have a different response to different actions by player 1.
The standard game theory assumptions are made: Players can rank their preferences, and all players know the preferences of all other players. With this in mind, a tree diagram may be drawn showing each possible action, and each final outcome. Below is an example involving a standard kidnapping scenario (pardon the pathetic ascii art):
RETURN ### (3,5)
CHILD ###
###
.###
###
####
###:
###,
###.
.##
####: ### KILL
#### .### CHILD
#### ###
.#### ###
PAY #### ###
RANSOM #### ###
#### ###
####: ####
#### ####
#### ### (-10,3)
####
#### NORMAL
WORRIED ;### KIDNAPPER
PARENTS ####
####,
#### # (5,-1)
#### ###
##### RETURN ###
;### CHILD ,###
####; ###
DON’T ,### ####
PAY ##### ####
RANSOM ;#### ###:
#### ###
.#### ###
#####
###
### KILL
### CHILD
###
###
###.
###,
###;
####
## (-5,0)
In this game the kidnapping has already taken place, and so the first move is made by the parents. They have two options: They may pay the ransom, represented by the upper branch, or they may refuse to pay, represented by the lower branch.
Once this decision has been made, we see that it's the kidnapper's turn. His name is in the center of the tree, at the point where his decisions will be made. He has two options: kill the child or return it to its family.
Note that at each possible outcome there are two numbers in parentheses. These are the payoffs that each player gets if this outcome is reached. The first payoff going to the first player, and the second to the second. So for example, if the ransom is paid and the child is returned, this is the best outcome for the kidnapper, who gets a payoff of 5. The parents get a payoff of 3 in this case.
So how do we solve this game? What's the final outcome?
Solving a sequential game uses a method known as backwards induction. Very simply, we start with the last player (the kidnapper), and map out his decisions. Then we move backwards to figure out what decision the parents will make.
Suppose that the parents pay the ransom. In this case, what would the kidnapper do? By looking at his payoffs we see that, because he is not a sadistic bastard, he has a higher payoff to returning the child (5) than he does for killing it (3). So if the ransom is paid, the child will be returned.
Now assume that the parents have not paid the ransom. What will the kidnapper do in this case? Here, we see that he has a higher payoff to killing the child (0), than to returning it (-1), perhaps because it will wound his ego, etc. So if the ransom is not paid, the child will be killed.
Finally, having mapped out all of the options for the kidnapper, we move to the parents. They know that if they pay the ransom the child will be returned, and if they do not it will be killed. While the best possible outcome for them would be for them not to pay the ransom and still have the child returned, it is impossible for them to ever achieve this. As a result, they only look at the two outcomes that are possible for their actions. In this case, that would be the outer two branches. The parents see that if they pay the ransom, they get a higher payoff (3), than if they do not (-5). So, the parents will pay the ransom, and the child returned.
This dynamic would be impossible to capture in a simultaneous game. The kidnapper's response depends on knowing the parents' action first.
Note that this outcome is solely dependent on the preferences of the players. We could run this same game with a sadistic kidnapper, who would kill the child no matter what. In this case, the best move for the parents will be to not pay. Their child will be dead, but at least they won't be out the money too.
I leave you with my favorite sequential game, demonstrating that poor old Charlie Brown should never try to kick the ball. He should know that, once he starts, Lucy will always get greater utility to pulling it away and making him look like a fool:
### (10,2)
LET ###
CHARLIE ###
KICK .###
BALL ###
####
###:
###,
###.
.##
####: ###
#### .### PULL
#### ### BALL
.#### ### AWAY
KICK #### ###
BALL #### ###
#### ###
####: ####
#### ####
#### ### (-10,10)
####
####
CHARLIE ;### LUCY
BROWN ####
####,
####
####
#####
;###
####;
DON’T ,###
KICK #####
BALL ;####
####
.####
### (0,0)