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Section 2d Game theory Game theory is a way of

thinking about situations where there is interaction between individuals or institutions. The parties may have conflicting interests. The theory is a structure to help us sort out our thinking in these situations.

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StrategyIn a game of tic-tac-toe your next move will be influenced by my move and my move will be influenced by your move.

A strategy is a plan for acting that responds to the reactions of others. Game theory deals with situations of strategy.

To characterize a game we must specify 3 things

1) the players, 2) the strategies of each player, and 3) the payoffs to each player for each strategy.

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Prisoner’s dilemmaA classic game is the prisoner’s dilemma. Here is an example, slightly different from the book.

Two career criminals are picked up for a recently committed crime (which they may or may not have committed) due to the circumstances of their recent activity. The two are taken to the police station and put in separate cells where they can not communicate with each other. Each is offered this deal. Confess and you get 1/2 year if the other does not confess, but you get 5 years if the other also confesses. If you don’t confess and the other does confess you get 7 years, but if the other does not confess we can get you for 1 year in jail.

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Strategic form of game

The information about the criminals on the previous page is typically put into a payoff matrix. One player is put in the rows of the matrix or table and one is put in the columns. In each cell we see the payoff for each player, given that outcome. A negative number represents a lose and a positive number represents a gain. The first number in each cell is the payoff to the row player and the second is the column player payoff.

Let’s put the information about our example into a table.

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Our examplecolumn player

confess keep quiet

row confess -5, -5 - 0.5, -7

player keep quiet -7, -0.5 -1, -1

The assumption here is that each player would like to have the highest number possible - in this case that means serve the lowest jail sentence for themselves.

Each player has to decide what to do in isolation. We will start with the row player.

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Row playerThe row player “sees” her options as the first number in each pair. Now she will say

If the column player confesses the best I can do is to confess - I get 5 years instead of 7.

If the column player does not confess the best I can do is confess - I get a half year instead of 1 year.

So the row player sees that no matter what the column player does it is best for her to confess. The row player is said to have a dominant strategy - here confess.

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Column player

The column player “sees” his options as the second number in each pair. Now he will say

If the row player confesses the best I can do is to confess - I get 5 years instead of 7.

If the row player does not confess the best I can do is confess - I get a half year instead of 1 year.

So the column player sees that no matter what the row player does it is best for him to confess. The column player is said to have a dominant strategy - here confess.

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Nash equilibriumIn this game each player will confess because that is the best each can do. So each will get 5 years.

Would there be any reason for either to change?

If each thought the other would not change, then there is no reason to change. Changing would just make the player worse off. This idea of no reason to change is called an equilibrium and here it is called a Nash equilibrium in honor of the guy who thought about this idea a lot.

Some games have no Nash Equil., some have more than 1.

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Pareto-efficientAgain I ask the question, would either want to change from the confession?

If each could be sure the other would not confess, they both would be better off not confessing. So the both confessing solution is not pareto-efficient because both can be better off with a move.

The problem here is could both trust the other to not confess? If the row player confesses, what would keep the column player from keeping quiet? The row player here will not risk cooperating with the column player, and vice versa.

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The super bowlSay that I have a special bowl in my possession. I can use it as a bowl, but also in exactly one year the bowl will unravel into 100 single dollar bills and the bowl will be gone.

Would you pay me $105 today for the bowl? How about $95.45, or $90 or 80? Would I accept all your offers? Well if the interest rate, r, is 10% then every dollar today can be made into $1.10 by next year --- 1 + 1(.1).

In general P dollars today can be thought of as F in one year, where F = P(1 + r).

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The super bowlYou would not pay $105 because next year, with the bowl, you would only have $100. But if you invested your $105 you would have $115.5.

If you paid 90.91 for the bowl you would have $100 with the bowl, but also $100 without the bowl. 90.91(1 + .1) = 100 (if you round to nearest penny). So you would not pay more than 90.91 for the bowl because you would be worse off at the end of the year.

Would I accept 80? If I took the $80 and invested at the interest rate I would have only $88. I would lose out if I accepted this trade. I would take exception to the trade.

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Present value of a future amount

The equation F = P(1 + r) means that if you have P dollars today and you can earn interest rate r, then you will have F at the end of one year.

You can change the equation to P = F / (1 + r) and say that if you want F dollars at the end of one year and you can earn interest rate r then you have to start with P dollars today.

If the time frame is longer than one year then you have to take the term (1 + r) and raise it to the power n, where n is the time frame.

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Basic formulasIf F and P are n years apart, then the relationship between F and P can be stated

F = P(1 + r)n or

P = F / (1 + r)n .

Sometimes will may see a stream of payment at various points in the future. The present value is simply the sum of the present value of each part.

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The theory of asset pricingAssets like our labor, or maybe an apartment building, can give us a future stream of payments. What are we entitled to if someone damages our asset and lowers the future stream of payments?

Typically we say the net present value of the damage to the future stream.