# Cheating for a \$20

Each year in my 500-student principles class I gather a group of eight students and tell them that I will auction a \$20 bill to the highest bidder. If two or more students bid the same thing, the difference between \$20 and their joint bid will be divided among the winning bidders. They can collude to fix the price just like oligopolists who violate antitrust laws, but they must mark down their bids in secret.

Today seven of the students stuck to the collusive agreement, and each bid \$.01. They figured they would split the \$20 eight ways, netting \$2.49 each. Ashley, bless her heart, broke the agreement, bid \$0.05, and collected \$19.95. The other 7 students booed her, but I got the class to join me in applauding her, as she was the only one who understood the game.

It showed that, even in a market like this one with very few players, collusion is difficult to maintain. There are tremendous incentives for one or more parties to cheat and move the market toward a competitive outcome. Unfortunately nobody has ever gone as high as the predicted equilibrium bid of \$17.50.

#### KW

Long live the Ashleys -they'll create jobs for our children.

#### Joey

I'm skipping my Game Theory class as I read this...

#### nate

why would the equilibrium bid be 17.50? i would think it would be 19.99

#### rda

Is being known as the "cheater" in a class of 500 worth \$20?

#### relaxing

The exercise only says the stakes weren't high enough for those colluding to engage in extortion or violence.

#### Silas Barta

No, she learned how to game a very narrow instance of that type of scenario, and got lucky that someone else didn't bid \$0.06.

Try that kind of thing in real life, and you'll get the social equivalent of a horse's head in your bed.

Incidentally, how many friends did Ashley make out of this event?

Well Ashley was able to do this because this is not a multi period game. Ashley would have cooperated if this was a multi period game and there was a different way where other seven players could harm her (in a different game may be). There was an absence of threat which made it possible for her to cheat.

#### Howard

"KW" you are oversimplifying. The pursuit of money does not always create jobs.

I for one will neither applaud nor deride Ashley. Greed is as much a part of the human condition as happiness/sadness.

#### Mike B

You do realize that game theory also demands that the people she cheated must retaliate so that she is incentivized to never cheat again.

#### ifidontmind

It would be interesting (though costly!) to see how the game would evolve if you played for many rounds. Would the bid approach an equilibrium bid of 17.50 (which I assume you got by dividing the \$20 by the number of players), would it equilibrate at the collusion bid of \$0.01, or might it oscillate between the two?

#### Richard B.

To profit most from this game, not only do you have to cheat, but you have to convince your competitors that you will not cheat. In other words, you have to be the most convincing liar to keep the bids low and profit the most.

#### DNS

I'm certainly not an economist, I don't think this applies that well to real-world antitrust issues. This is a one-time competition for a fixed prize. In most real-world situations that I can imagine, the competition has a chance to react.

Attempts to compete by one party, i.e. by lowering prices on a good or service, would result in only a short-term gain. Their competitors would quickly realize that the agreement had been broken, and initiate a price war, resulting in mostly the same competitive landscape, but lower profit margins for everyone.

I think a more accurate experiment would be to use the classic \$20 open auction, but add the split-on-tie rule. Perhaps set a 5-cent bid increment, to better simulate the nature of the price war. With a limited number of intelligent players, my guess is that, even if forbidden to fix the price, they would converge within a few rounds.

#### Will Cross

She also has an upper limit on the value of her reputation for honesty: \$17.46 (the net gain she got from her betrayal of the group).

As for the chumps,

I'm willing to bet that this is one reason why the value stays below \$17.50: any time an agreement is reached, breaking it means upsetting the other 7 (or perhaps fewer, if there are multiple cheaters). Collusion drives the price down to nearly 0. Also, if you played this multiple times (with different groups of eight, without overlap between groups) you might see that equilibrium value go up towards \$17.50 as people learned to game the system.

#### MM

I would have bid 20.01\$. Why? Because I can.

#### tim

@ nate: I agree, I'm trying to come up with the \$17.50, but I can't. It seems like a typical prisoner's dilemna type game to me. Clearly winning \$2.50 is better than winning \$2.49 but if you expect others to cheat as well then you're better off bidding up the price to \$19.99 to at least take away something.

#### Brian

And what would be the typical outcome when the game was played with the same 8 players repeatedly. It is one thing to point out the behaviour of repeated trials using only first time players in each trial.

But if you used the same 8 players repeatedly they would learn behavior with eachother. Maybe they would never reach an equilibrium of total collusion, but it would eventually break down to large collusion and an occasional non-participant.

Also I would prefer if the players had to actually give up what they bid (I doubt you collected 12 cents from 8 people) after repeated bids. I bet the average bid price would rise, but the frequency of collusion would also rise in response, after people (companies) got tired of being burned.

#### Phil Scott

nate, I was thinking the same thing. But the highest I would bid would be \$2.50. Worst case scenario everyone bids the same as me and then I'm out nothing (I spent \$2.50 for \$2.50).

If you went with \$19.99 and someone else did the same, then you two would be bidding \$19.99 for \$10

#### mike

Long live the Ashleys?--They'll create jobs for children (literally) overseas. Yay capitalism and it's eternal quest for the lowest common denominator. Aren't we proud that humans can't do any better.

#### Zoltan

A handful of commenters deride Ashley, or say she has ruined her reputation. Perhaps not - perhaps she's earned the rep. as someone who understands what the teacher wants, understands how to get things down, and is clever. I would have done the same thing.

#### dd

I don't see how she cheated. She played the game and won. I'm surprised she's not getting more support here in the land of individualism and healthy competition.