Improspectives

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Archive for the ‘Game Theory’ Category

Prisoner’s Dilemma, Part 1

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The Prisoner’s Dilemma is perhaps the classic 2 x 2 game. The scenario is simple enough to explain, but it seems impossible to find a way out of the dilemma. Here’s the situation: You and a fellow criminal have been apprehended, and the police want at least one of you to give evidence against the other guy. They tell you that if neither of you talks, they have enough evidence to put each of you in jail for a relatively short time. On the other hand, if you give up the other guy and he refuses to talk, he will be convicted and sentenced to a long term, and you will go free. Of course, if you don’t talk and he does, the same thing happens to you. If you both talk, you will each get a sentence that’s worse than you would get if you were the only one to defect but not as bad as when you didn’t defect and your partner in crime did.

You can summarize the Prisoner’s Dilemma payoffs using the following 2 x 2 grid.

Graphic showing the payoff grid for the Prisoner's Dilemma.

The central question is: What is the best way to play this game?

Obviously, it’s in your best interest to cooperate. If both you and the other person cooperate (that is, you cooperate with each other and don’t talk), you will get a sentence of only one year and minimize the negative payoff. The problem is that if the other individual knows that you are going to cooperate, he has no incentive to play along. He should defect (turn you in and get away with no jail time at all). To avoid the possibility of a longer jail term, you should also turn in the other person, giving him a medium sentence and assuring that you don’t get the longest possible term.

So the strategy for playing the Prisoner’s Dilemma exactly once is to always defect. You don’t get the best possible payoff, but you do prevent yourself from getting the worst possible outcome. The same considerations work for improv and business. If you never plan on performing or working with someone again, what they think of you and what they might do to you in the future is irrelevant. You could choose to defect by breaking a promise or paying an invoice late and move on with your life knowing that the other individual or business won’t be able to exact revenge. But what if you play game multiple times? That’s what’s called an Iterated Prisoner’s Dilemma.

Next time: the Prisoner’s Dilemma and ongoing relationships.

Stag Hunt Across Species

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In the Stag Hunt, also known as the Assurance Game or the Coordination Game, two hunters have the option of going after a hare or a stag. If a hunter targets a hare, he will get one for a small payoff every day. If both hunters go after a stag, they will each receive a much larger payoff, but if you target a stag and the other hunter goes after a hare, you will get nothing. Your choice as a player is whether to go for the reliable but small payoff or the larger, riskier payoff.

As with most of the other classic 2 x 2 games, we assume that the two hunters can’t communicate. At least, not so that they can coordinate their efforts before they choose which strategy to follow on a given day. What they can see, however, are the payoffs both for themselves and for the other hunter after they play a round of the game. This situation leads to some very interesting outcomes, especially when you consider it across species.

The authors of a paper titled “Responses to the Assurance Game in Monkeys, Apes, and Humans,” using equivalent procedures, tested whether capuchin monkeys, chimpanzees, or humans would perform better at the Assurance Game. The research team chose these three sets of primates because they all show tendencies toward cooperation in their natural environment. What made their research interesting was that they did not allow the humans to talk while they were being tested.

As you might expect, the humans performed better at the task than either the chimpanzees or the capuchin monkeys, but the difference was not as large as one might expect. Specifically, the researchers found that the humans who discovered the hare-hare approach thought that they had beaten the game and were always getting a reliable payoff. Game theorists call this approach the risk dominant strategy. Once the players achieve a reliable outcome, they tend not to move away from it and explore alternate possibilities.

The paper’s authors summarized their results this way:

Finally, despite being the species for which the highest frequency of pairs achieved the payoff dominant outcome, even among humans fewer than 20% in pairs did so (this increases to 27% when borderline pairs are included). An additional 38% of pairs achieved the risk dominant outcome (hare-hare), and 12% matched their partner. It is worth reiterating that despite success of humans compared with the other primates, a nontrivial proportion of pairs failed to achieve the payoff-dominant outcome. This underscores the difficulty of finding outcomes when the typical human procedures (instructions, payoff matrices, pretest for understanding) are absent, common handicaps for nonhuman species.

Obtaining the highest possible payoff from any venture, whether it is in improv or in business, means taking risks. Even in this artificial situation, in which communication was limited, some pairs of humans managed to find the payoff-dominant outcome for the Stag Hunt experiment. The problem was that many of them did not. As the authors of the study note, this is most likely a case of humans being risk averse. In an improv context, being risk averse might mean always asking for the same type of suggestions or doing the same type of scene, regardless of which suggestion you get. Some so-called improv groups even get a single suggestion from the audience, use it once by stating it during their scene, and then do the rest of the scene according to a script. It’s a cheat, one that takes a lot of the fun out of doing your performances unless you change the script every night, but it does reduce the risk of having something terribly wrong and not entertaining your audience. Over time, however, taking larger risks will yield greater rewards as long as you have competent individuals on the stage with you.

Coordinating to Win Big in the Stag Hunt

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This series of posts talks about game theory and applications of the classic 2 x 2 games to improv and business. I’d previously discussed Chicken and Battle of the Sexes–next up is the Stag Hunt.

The Stag Hunt game is also called the Assurance Game or the Coordination Game. The basic idea behind the Stag Hunt is that the players can increase their payoff by cooperating. Following one traditional way to state the game’s conditions, I’ll define cooperation as a situation in which two individuals can increase their immediate payoffs by working together.

In its traditional form, the Stag Hunt posits that there are two hunters. The hunters can either choose to hunt for a stag, which has a very high but uncertain payoff, or a hare, which has a small and reliable payoff. Every time you hunt for a hare you will catch one and get a small payoff. On the other hand, if you hunt for stag and the other hunter goes after a hare, you will get nothing, and the other hunter will get the hare. You get a stag only if both of you decide to go after it. That’s why this game is also called the Coordination Game: You must coordinate your efforts to get the highest possible return.

The following matrix displays the payoffs for the Stag Hunt.

A graphic showing the payoff matrix for the Stag Hunt game.

As with most of the other classic 2 x 2 games, we assume that the two hunters can’t communicate. At least, not so that they can coordinate their efforts before they choose which strategy to follow on a given day. What they can see, however, are the payoffs both for themselves and for the other hunter after they play a round of the game.

Obviously the best possible outcome is for you and the other player to go for the stag every round, but adopting that strategy is difficult when you don’t know the payoffs in advance and can’t communicate during the game. The best solution, both in business and in improv, is to find a way to communicate, even if it means making taking several moves to try different choices and observe the payoffs. Then, if you make a choice that leads to the big payoff, you trust that the other player will cooperate and repeat the move.

Next time: Do humans play this game better than chimpanzees?

Battle of the Sexes

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In my previous two posts, I discussed the classic game of Chicken, where the loser is the first person to give way to the other. Another classic game is the Battle of the Sexes, which imagines a situation in which neither player can get their preferred outcome, but the worst possible outcome is disagreeing. Let’s say you and your partner are invited to a party, and the host has asked you to bring beer or wine, but not both. Now also assume that you can’t get in contact with your partner. It’s the lack of coordination that makes these 2 x 2 games interesting and aggravating.

For the sake of argument, let’s assume that you are a beer drinker and your partner prefers wine. For you, the best possible outcome is if both you and your partner decide to bring beer. The second best outcome is if both of you decide to bring wine, and tied for the worst are when you and your partner bring different beverages. The problem is that you have no way to decide whether to bring beer or wine. If you base your decision on your partner’s preferences, you will bring wine. On the other hand, if you think that your partner will go along with your choice, you will bring beer. There’s no way to place one of those two options over the other. What’s worse is that your partner has exactly the same problem.

In terms of improv, you’ll find the Battle of the Sexes occurring when there is a lack of communication before you start playing a scene or game. Every improv group plays games differently, even when they’re based on the same familiar pattern. If you’re playing in the city as a guest player, there might be two different ways of playing the same game. If there is a lack of communication, such as if you go over to one side of the stage to pick up a costume piece before conferring with your playing partners, whoever starts might not have the same idea about how to play the game as everyone else. If your scene partners play one game, and you’re playing another, a train wreck ensues. It’s easy enough to fix once everything gets underway – either you or your playing partners can adapt, but there might be an awkward moment or two at the start. At that point, you just hope the audience either doesn’t notice or forgives you.

In business, you’ll find that the Battle of the Sexes game is played out during sales calls and engineering meetings. Everyone has a preferred solution for implementing a change or creating a product. Any time there are multiple pathways to creating a product or finishing a project, you should be in close communication to ensure that the solution you’re pursuing doesn’t contradict what someone else is doing.

Chicken in Business

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In my previous post, I discussed the game of Chicken, where two kids in denim jackets and James Dean haircuts get into cars and drive toward each other at high speed. The first one to swerve loses, but there’s a real possibility neither one will blink and both will be hurt. If you’re an improviser, it often pays to be the one who blinks. Yes, you go with someone else’s ideas instead of your own, but the scene and show will be better for it.

In business, one of the most common ways to play Chicken is what’s called Schedule Chicken. In Schedule Chicken, managers face off against each other in a meeting room and none of them is willing to admit that they will not meet whatever obviously unrealistic deadline has been placed in front of them. Because they agreed to that schedule at the start of the project, whoever blinks will be blamed for causing the project to slip if they have to ask for more time.

Unrealistic schedules are deadly. In her Harvard Business Review article “How to Kill Creativity”, Teresa M. Amibile notes:

Organizations routinely kill creativity with fake deadlines or impossibly tight ones. The former create distrust and the latter cause burnout.

In business, it can be tough to say that you have no chance of meeting a schedule, regardless of how optimistic it might be. If the other managers or workers on a team say that they can make their deadlines with no problem, it means that you are the one causing the slip. Of course, it might be complete fiction that the other teams could have been ready in time, but if you’re the first one to admit that you won’t make it, you’re the one who gets the blame.

How you solve the game of Schedule Chicken depends on your corporate culture. Companies that follow a philosophy of not shipping until something is ready can reduce the possibility of Schedule Chicken, especially if they don’t set final deadlines until the project is well underway. For multiyear efforts, final deadlines and announcements should be kept out of the press as long as possible. Companies that use agile development and roll out small updates frequently avoid Schedule Chicken by shipping when the update is ready and not announcing times until the next increment is ready to go.

Written by curtisfrye

September 20, 2012 at 1:02 pm

Improv and the Game of Chicken

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If you’ve watched any movies from the 1950s about disaffected youths, you are surely familiar with the game of Chicken. In the game of Chicken, two kids drive toward each other at high speed. The first person to swerve loses – that is, that person is the chicken. As with all 2 x 2 games, there are four ways for it to play out. The payoffs appear in the following table.

Image

The first outcome is if both drivers swerve, which results in payoffs of zero. Both drivers swerve, so that means that neither of them won. But, because the outcome was equal, neither of them lost, either. The next two outcomes occur when one driver swerves and the other stays straight. In that case, we do have a clear winner and a clear loser, which is reflected in the payoffs of plus one for the winner and minus one for the loser. In the fourth case, disaster strikes. In that fourth case, neither driver swerves and there is a high-speed, head-on collision.

You probably don’t have to stretch your imagination to see how this game can play out in improv and business. When you create an improv scene, someone has to be willing to give up control. Even if it’s only for a moment, players must accept what other players say and do so they can continue to build a consistent reality without interrupting the audience’s enjoyment. The best outcome in a game of Chicken when you’re performing an improv scene is to have one player swerve and one player continue straight on. That means one player made a solid decision and all the other player has to do is follow along and build on what is been established. If both players swerve, that means no one is taking the reins and attempting to drive the scene forward.

Next…chicken and business.