Economics micro learn: Game theory and pay off tables

Game theory is the idea that every activity we do, or decision we make can be thought of as a game. A game may be simple or complex and may have strategy that is obvious or subtle.

If you remember from our last session we were cutting cakes.

cake 50 50 60 40

There are only two ways to cut a cake really, fairly or unfairly. The way the cake is cut depends on the rules of the game.

Let’s make the rules of the game.

There are two players.

The first player cuts the cake into two parts, each part may be any size.

The second player chooses one of the two parts to keep as their piece of cake.

The first player then keeps the part of the cake the second player did not take.

* I think we can all agree that if player 1 chose which piece before player 2, player 2 would frequently receive nothing.

Most of us learned this game as children, usually a parental figure gives two children a food item to divide and states the rules similar to the way I have described them.

Usually this game results in fair distribution if the rules are followed.

You should be able to see in your mind how this game generally plays out. However there are circumstances where a very different result occurs.

To explain this I will use a pay off table.

Player 1 has two possible strategies 1) be fair OR  2) not be fair  in the cutting of the cake.

Player 2 has 2 choices 1) be smart and take the biggest piece OR 2) not be smart and take the smaller piece.

* It is possible that player 2 is not hungry, or is unable to determine which piece is larger (49% vs 51%) or feeling quite generous, or is a woman on a first date.

When playing these games we assume the opponent is rational and is seeking the best possible outcome for themselves. If you opponent is not rational it is difficult to apply in many cases.

Generally in game theory the following convention is used, player  1 uses rows (horizontal)  to determine their strategy and player 2 uses columns (vertical) to determine their strategy.

The result of the game is where the two strategies intersect.

For each strategy combination there is the corresponding box in a grid with 2 numbers  separated by a coma, the number on the left is players 1 reward, the number on the right is player 2 reward.

Reward numbers are positive for a gain and negative for a loss.

In the pay off chart below I have color coded the rewards to match the players, though this is not commonly done. Think of this as economics paint by numbers.

We can use the pay off chart to analyze the game.

Intro to game theory

Lets assume I am player 1

The first this you should notice is if I cut the cake fairly I know what my reward will be, and it doesn’t not matter if my friend is smart or not.

The second this you should notice is the highest and lowest rewards in the game for me occur when I cut the cake unfairly. Cutting the cake unfairly is therefore a risky strategy and I am gambling on my ‘friend’ being stupid in order to get a larger reward.

Thirdly because the game seems have an unofficial title of “Are you stupid my friend?” a game like this can be used to test the intelligence of an opponent.

Fourthly an opponent in this game can ‘play stupid’ and let you take the large reward assuming you will assume he (or she) is stupid in the future when you risk even more on a future game ( 75 vs 25).

cake 2

Player 1   Player 2

Cake 1       60                40       ( Player 1 unfair, player 2 not smart (apparently…) )

Cake  2      25                75       ( Player 1 even more unfair, player 2 suddenly smart ( surprise!) )

Totals        85               115      ( Who is stupid now?)

This type of strategy is the conman game or poker game.

Interestingly the conman game strategy is almost foolproof as long as a second game follows and player 1 is still unfair. The only way a prospective conman game can loose here is if player one switches strategy to fair (or less unfair, 55 vs 45) after their initial windfall. Ironically the conman can be defeated by player 1 getting a conscience or giving up the gamble.

The other important thing about games is sometimes the same games are repeated and sometimes they are played with different games in between and sometimes games are only played once. Entirely different strategies apply to a series of games as apply to ‘one off’ games.

Who thought cutting a cake into two parts could be so complex?

This is why I get very concerned when people talk about ‘redistribution of wealth’.

If all of this was too difficult for you to understand I have some sound advice for you, try to be fair in your dealings, and never trust a socialist, a banker, a salesperson or a anyone employed by the government.

We put the NOM in economics!

Posted on April 20, 2013, in Economics & Finance, Education, Physical resources and tagged . Bookmark the permalink. 3 Comments.

  1. So a woman on a first date takes the smaller piece? LOL I swear, if you get a bunch of women dividing up a cake amongst each other, they will all fight for the smallest piece.

    • The ‘woman on a first date’ scenario I was describing was a romance themed first date. Isn’t it interesting that when women are involved in a non romance cake dividing game it is suddenly very different. A completely different game.

  2. I should point out that the player choosing second faces a thing called ‘adverse selection’ which is making a selection which is not under the best conditions. This assumes the first player to choose takes a piece, trying to take the largest piece. The second player often get the crumbs if the first player cuts the cake and chooses first.

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