The Ultimatum Game and The Hollywood Writer’s Strike

Once during college, an annoying person asked me for game theory help. I offered to help even though it interrupted my own work. It was hard to turn down someone who lived in the same dorm.

I offered to help, but wanted something in exchange. The person was a teaching assistant for a non-credit class my friends were taking. He was essentially using his power to fulfill a personal grudge, so I asked that he be nicer to my friends and act more professionally. He was furious at my suggestion since he felt it undermined his authority.

So I said I would not help him. I reminded him that I had work to do, so if he could not compensate me for my time, I would resume my studies.

He angrily responded that he didn’t need my help after all. He actually knew how to do the problem, he indicated, but he just wanted my perspective. And, he continued, the professor was holding office hours the next day, so he would ask someone smarter.

He then walked away, sort of.

As I resumed my studies, he conspicuously sat down at a neighboring table and faked that he was doing work. Every few minutes he peeked over his book to see if I would budge.

But I didn’t.

After thirty minutes, someone who saw the whole thing came up to me and said, “Presh, I can’t believe you aren’t helping him with game theory. I mean, I know you don’t like him, but come on. He needs your help.”

To which I replied, “I am helping him. If he knew any thing about game theory, he would know that I’m not going to tutor him unless he concedes to my request.”

On hearing this, the annoying person left the room and never asked me for help again.

I guess he did not really need my help after all.

I chose my actions because I had total negotiating power–a concept from the ultimatum game.

I am the decider!

Imagine you are participating in a college experiment. The researcher hands you $10. You are instructed to propose a split of the money to another person. For instance, you might propose that you keep $7 and the other person gets $3.

If the other person accepts the division, the money is split according to the proposal. If the other person rejects the division, neither person gets anything.

How might you go about deciding a split?

The first thing you can do is diagram the game dynamics:

You are player one (black) and will offer money to player two (blue), from $0.01-$10. After hearing the offer, the second player can either accept so the split goes through, or reject and leave both parties with nothing.

The way to analyze the game is to start at the end of the diagram and think about what the second player’s strategy. This is a technique called backwards induction.

What offers will player two accept?

If he rejects an offer, then he goes home with nothing. If he cares about getting money, this would suggest that player two’s strategy is to accept any offer that player one makes. Even if the offer is for one cent, that’s better monetarily than going home with nothing.

In the land of infinite rationality, player one understands this, and takes advantage of his bargaining power. Knowing any offer will be accepted, player one will offer the smallest amount to player two–one cent–and basically take the entire pot for himself.

This solution–player one gets everything and player two gets nothing–is why the game is called the “Ultimatum game.” Player one has complete negotiating power and knows any ultimatum will be accepted.

But doesn’t player two have the power to punish?

The situation stinks for player two. It just does not seem fair that player one gets everything. I mean, player two does have the ability to punish player one by rejecting offers. Can that lessen player one’s power?

Imagine player two tries to gain power. Before the game starts, player two might try to taunt to player one, “Hey, if you offer me less than a 50-50 split, I will reject it. I don’t care if I go home with nothing; I’ll just be happy to know that I hurt you.”

In the theoretical world, player one understands this is a non-credible threat. If player one does offer a smaller amount, then player two faces a one-time decision of some money or no money. In this world, it only matters that one cent is better than nothing. There is no joy in inflicting pain on the other player. And because the game is not repeated, there is no reason player two should even care about punishing player one.

As a technical aside, each offer that player one makes actually is a “subgame” where player two has to make a decision. Since player one always offers one cent and player two always accepts, we never observe what would happen if player one offers two cents or more.

In theory, player two could do whatever he wants–he could reject those higher offers. But as argued above, the only credible thing player two would do in any subgame is to accept the offer.

So we call the equilibrium where player two accepts any offer and player one gets everything the subgame perfect equilibrium. In this game, it is the unique subgame perfect equilibrium.

But Presh, that’s simply not what happens in real life!

Yes, imagine you are playing this game with your crazy girlfriend, and she’s player two. From experience, you know that she might even reject offers where you give her less than 90% of the money.

You’d be wise to forget the subgame perfect equilibrium stuff and just make sure you keep her happy. She’s crazy after all.

Indeed, experimenters have repeatedly shown that the subgame perfect equilibrium does not happen. Here’s one example.

But I believe these experiments do not mean game theory is wrong. In fact, they illustrate why game theory is a powerful tool.

It’s all about the beliefs, man

A Nash equilibrium is based on your beliefs about other players, and how accurate those beliefs are. I gave some intuition about the Nash equilibrium in this post.

The subgame perfect equilibrium is the only Nash equilibrium where both players have infinite reasoning ability and they only care about money.

What people seem to miss is that the ultimatum game really has an infinite number of Nash equilibriums.

Let’s go back to the game with you and your crazy girlfriend. Your strategies are something like the following:

Her: “Give me at least 90%, or I reject.”

You: “I will make sure she does not reject. I surely want to leave with some money.”

If you really believe her threat, and it is accurate, the resulting division is a Nash equilibrium of the game. She is not playing rationally, but the result of you getting $1 and she getting $9 is in fact a Nash equilibrium.

Her threat to reject should be non-credible, except that she cares more about money. She cares about power, fairness, and yeah, she’s crazy.

Ironically, there is power in irrationality.

The different outcomes of the ultimatum game can illustrate how beliefs differ across societies. Americans often reject offers less than 20%. My Israeli game theory professor was proud to note that Israel is one of the few places where low offers were given and accepted.

Here’s an article with more details about the cross-cultural differences (.doc file)

But when does this game happen in real life?

The game happens all the time in homes. My brother and I would be arguing about something, like what video game to play. Eventually, my parents would tell us that we could either agree on a game or we would not get to play at all. My older and stronger brother would often speak first and pick a game that only he would like. And I was stuck with the game because it was still better than nothing.

A similar, though not exactly the same situation, happens with labor strikes. The boss offers terms to workers. If the workers accept, then both workers and the company profit. If the workers reject and go on strike, both sides get nothing and lose money while waiting.

In the current Hollywood strike, both the writers and producers are losing. The cancelled Golden Globes is estimated to be a $75-million to $100-million loss to the L.A. economy.

While the strike persists, each side will try to gather negotiating power by tricking the other side on what it truly believes. Each side will want to demonstrate to the other that it will reject solutions that are deemed unfair, even if that action appears crazy.

As analyzed on the blog of James D. Miller, the producers are showing they are committed to a prolonged strike because they have Rupert Murdoch on their side:

According to Slate.com the producers are following the hard line of Rupert Murdoch. And, as Slate.com points out, Murdoch might benefit from a strike because his main cash cow, American Idol, is unaffected by the strike, whereas American Idol’s scripted competition will be forced to show reruns if the strike continues.

Slate.com gives the following warning to producers “Does it occur to the other networks, which have no such weapon in their arsenal, that by following Murdoch’s hard line, they are enabling him to get the most out of his competitive advantage while they are left to scramble with reality shows and reruns of cable programming?”

My bet is that not only do the other networks realize this but they consider it a good thing! If the writers come to believe that Murdoch wants the strike to continue and that the other producers are going to follow Murdoch then the writers will realize that the only hope of avoiding a long strike (i.e. the car crash) is for them to give in.

In response, I think the writers are drumming up what their beliefs really are. They want us to think this is all about socialism, and that they will not stop until a just solution is met. Even if the writers are not devoted to socialism, it could help their negotiating power.

Who will win?

I can’t say, but when you see who does win, it will be clear what each side believed. And it will be clear who had more power.