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Entries in prisoners dilemma (1)

Thursday
Feb032011

Game Theory

This photo taken for free (minus the guard's scowling from across the room) at Portland Art Museum's exhibit "Disquieted."

Coordination, Battle of the Sexes, Chicken, Prisoners' Dilemma: These are the four basic games in game theory. They compare how two players make a choice.

In game theory, payoffs for one player depend on the choices of the other player. Numbers represent the payoffs one player receives for himself relative to what the other player receives for himself; the numbers have no meaning beyond that relative comparison.

Players are assumed to be self-interested, meaning they want to get the highest payoff for any given decision. Also, the players make choices at the same time, so they can't conspire and work out what they plan on doing beforehand. 

1. Coordination Game
You just met Jo. You are taking Jo to Fancy’s on a first date. You and Jo both prefer to dress up formally, but either of you is willing to dress casually if that means you will match each other.

Neither of you wants to go formal if the other is going casual. Neither of you wants to go casual if the other is going formal.

The payoffs for each situation are assigned values in the table as two relative numbers: (You, Jo). Higher is better.

Coordination

 

2. Battle of the Sexes Game
After a few dates, you both decide your favorite place is Mellow’s. Now, you want to start dressing casually, but Jo still wants to dress formally. Again, you’d rather match than not match, either way.

Battle of the Sexes

 

3. Chicken Game
You and Jo break up. (Stupid clothing arguments!) You want to go have a drink at your favorite place, Mellow’s, but you don’t want to go there if Jo is going to be there. That’s the worst thing that could happen. (Jo thinks so, too!)

Neither wants to be the one to go someplace else and have the other get to keep going to Mellow’s. Both of you would like to keep going to Mellow’s, and have the other find a new place.

Chicken

 

4. Prisoners’ Dilemma Game
You can’t stand it. Why won’t Jo find a new place? That’s still what you want.

But you’d rather go to Mellow’s and scowl across the bar at Jo than go someplace else knowing Jo gets to enjoy Mellow’s free and clear.

And Jo feels exactly the same. (You two are made for each other.)

Prisoners' Dilemma

Why The Dilemma?
The prisoners’ dilemma is the most interesting case of the four. It’s the only situation where you both lose as a result of following your own self-interest. Both you and Jo are destined to lose when things get to prisoners’ dilemma stage. The decisions could play out like this:

In Coordination, times are good, and you and Jo both want the same thing. So if you think Jo will dress formally, you dress formally (because 2>0, and 0 is what you get if Jo goes formal and you go casual). If you think Jo will dress casually, you dress casually (1>0). Jo’s decisions and payoffs are the same.

Coordination

In fact, there’s no reason Jo wouldn’t choose to dress formally and get a 2, unless she thinks you don’t want to, in which case you are not in Coordination at all, but rather …

Battle of the Sexes, which is the same as Coordination, except one of you is going to lose slightly when the other gets what he or she wants.

If you think Jo will dress formally, you still dress formally (1>0), though Jo will enjoy it more (2>1). Likewise, if you think Jo will dress casually, you still dress casually (2>0), and you enjoy it more than Jo does (2>1).

Battle of the Sexes

Either way, you both want to match, because not matching means zeros. If you didn’t care about matching anymore, you and Jo’s relationship would result in a breakup and become a game of …

Chicken, which takes a different turn. Think of two cars headed toward one another on a one-lane road. If both do not swerve, they both crash and get zeros. In Chicken, no one wants to crash.

So if you think Jo will go to your old favorite place Mellow’s, you go someplace else (1>0). If you think Jo will go someplace else, you go to Mellow’s (3>2). In fact, if you want to be sure you avoid a zero, your best choice is to always go someplace else, because you will at least get a 1 and you might get a 2.

Chicken

The problem with going someplace else is that you really want to go to Mellow's. And Jo can win big if she knows you will always go someplace else, because she can go to Mellow's free and clear. That starts to bother you, making you jealous of her always getting a 3 to your 1. When you can’t get over it, the situation becomes a …

Prisoners’ Dilemma, where you are both trapped with a low payoff. If you think Jo will go to Mellow’s, you go to Mellow’s (1>0). If you think Jo will go someplace else, you still go to Mellow’s (3>2). And Jo’s payoffs are exactly the same to her.

Prisoners' Dilemma

So what happens is you both go to Mellow’s, scowl at each other all night, and get a 1—even though you both would get a 2 and be better off if you just went someplace else. (Ain’t it always the way?!)

 

"Facts of Life" image from Hulu

If You Play The Record Backwards
Maybe that's being too cynical. If we could run the story with Jo backwards, it might go like this instead:

Say you are in a prisoners' dilemma when you first see Jo at Mellow's: You're both in relationships with other people, and you both know going to Mellow's could tempt you to cheat, because you have seen each other several times and know you are attracted to one another.

After enough nights of winking (instead of "scowling") across the bar at Mellow's, you play a game of chicken and talk to one another. Both of you swerve, and you start a relationship at "a different place." (It's qismat! You both simultaneously picked the same different place. "Fancy seeing you here.")

At first, it's a battle of the sexes, where you're trying to decide who wants what and who likes what. But then you fall in love, and you both realize you want the same things. You're now coordinated, like the matching Adidas sweatsuits your aunt and uncle used to wear at family dinners.

Music break ...

Self-Interested Behavior is Inefficient
Unless you and Jo can stay together in your desire to match—or if you can’t cooperate after you no longer want to match—the prisoners’ dilemma seems inevitable. And it’s most inefficient, because the lowest possible combined payoff is guaranteed: 1+1 = 2, which is less than the other two possibilities, 3+0 = 3 or 2+2 = 4. (Again, the actual numbers are irrelevant, beyond a relative comparison.)

Why can’t you both go someplace else and leave Mellow's behind? Because neither of you can get past the mindset that says you must always do what gives you the most satisfaction: When you always act in your own self-interest, you are doomed to always go to Mellow’s and scowl (as long as it remains your favorite).

To the other player, it appears you've deliberately chosen lower satisfaction in order to inflict extra low satisfaction on them. Jo says, “Why would you come here and ruin both our nights, instead of going someplace else and both of us having a good time?” She can’t recognize the reverse schadenfreude: Each of you gets the most displeasure (0) when the other is most fortunate (3).

Fairness
This reverse schadenfreude occurs in money-sharing experiments, too, where people are generally happy to receive $1 for nothing, unless they know the person giving them $1 was given $100, had to give at least $1 of it away (based on the rules of the experiment), and kept the maximum amount ($99) for himself.

When experimenters allow the recipient the option of killing the entire deal—where no one gets any money—the recipient will do so, and receive nothing in exchange for the other player receiving a “bigger” nothing. The receiver will continue to kill the deal until his portion gets above $20, or so.

This behavior is an evolutionary survival strategy—a concept of “fairness” in the face of unearned gains that seems to be common to all human beings.

Photo "Direction of Time" by the author

Without Fairness, The Best Choice is “I Don’t Care”
If a recipient does not have the option to kill the deal, then his only choice is to be happy with $1 (or whatever he is given). Period.

But don’t be too cynical about it, because some few people will choose to split the money $50/$50, and even some tiny number of people will opt for a $99/$1 split in your favor. Experiment proves this does happen. (Thank you, Mother Teresa!)

More likely, the only escape from a prisoners’ dilemma is not to give a damn about payoffs at all. Because if you are truly powerless to cooperate and fairness is entirely out the window, you can argue and protest as much as you like, but changing the game is outside your control.

It is in your control to give the other person Mellow’s and the 100 bucks and wipe your hands of the situation. Your new price is zero. Your new game is freedom. Imagination. You swerve out of the way every time. You walk away happy … as long as you never, ever, never look back. And as long as you don’t mind being a “chicken.” BwooOOOooock, booock, booock, booock, boock, bock.

Playground tactics still keep us in the game.

 

Source for game basics: "Games People Play," Scott Stevens