Today we will take one last look at solving sequential games. Before we do that though let’s solve the example at the end of the previous blog.

Let p represent the probability that your boss writes you up and 1-p be the probability that your boss does not write you up. The probability distribution would be -10p + 40(1-p) = 20.

-10p + 40 – 40p = 20 =>

-50p = -20 =>

p = -20/-50 = 2/5

1-p = 3/5

So you will choose to shirk if the probability of your boss writing you up is less than 2/5. Or you will choose to shirt if the probability of your boss not writing you up is greater than 3/5. Of course you would never shirk to begin with and always work hard right?

In most of our examples we have looked at sequential games that involved two players. In the following example there will be four participants. In particular, Yogurtland is looking to establish a shop in a new market. However, three potential competitors (Mickey’s Yogurt, Yum Yum Yogurt, and Big Kahuna) are also debating whether to enter the market or not.

The payoffs are organized from top to bottom representing Yogurtland, Yum Yum, Mickey’s, and Big Kahuna respectively. We can solve this game using backwards induction. Let’s start with the options for Big Kahuna.

Scenarios for Big Kahuna (a lot):

Enter, Enter, Enter, Enter = 0

**Enter, Enter, Enter, No = 1**

**Enter, Enter, No, Enter = 3**

Enter, Enter, No, No = 1

**Enter, No, Enter, Enter = 3**

Enter, No, Enter, No = 2

Enter, No, No, Enter = 3

**Enter, No, No, No = 4 **

**No, No, Enter, Enter = 4**

**No, No, Enter, No = 4**

**No, No, No, Enter = 3**

No, No, No, No = 0

No, Enter, Enter, Enter = 2

**No, Enter, Enter, No = 4**

**No, Enter, No, Enter = 3**

No, Enter, No, No = 1

The preferences of Big Kahuna have been highlighted in red based on the payouts from each outcome. I also placed in bold the preferred actions of Big Kahuna above. Now let’s look at Mickey’s. Since Mickey’s assumes Big Kahuna will make optimal decisions, Mickey’s options are the following:

**Enter, Enter, Enter, No = 3**

Enter, Enter, No, Enter = 1

**Enter, No, Enter, Enter = 2**

Enter, No, No, No = 1

No, No, Enter, Enter = 1

No, No, Enter, No = 1

**No, No, No, Enter = 2**

**No, Enter, Enter, No = 1**

No, Enter, No, Enter = 0

Using backwards induction, Mickey’s optimal choices will be in blue. Let’s continue the process by identifying Yum Yum’s options.

Scenarios for Yum Yum:

Enter, Enter, Enter, No = 2

**Enter, No, Enter, Enter = 3**

No, No, No, Enter = 1

**No, Enter, Enter, No = 3**

Yum Yum’s optimal choices are now highlighted in green. As you can see, the options for yogurt shops becomes less as we use backwards induction. This leaves Yogurtland with a decision to make, enter or do not enter the market.

Their scenarios are:

Enter, No, Enter, Enter = 2

No, Enter, Enter, No = 2

Based on Yogurtland’s options, the company is indifferent between entering the market or not entering the market since their payoffs will be the same. So the solution to this game is (2,3,2,3) and (2,3,1,4) or (Enter, No, Enter, Enter) and (No, Enter, Enter, No).