Game Theory As A Dark Art
One of the most charming features of game theory is the almost limitless depths of evil to which it can sink.
Your garden-variety evils act against your values. Your better class of evil, like Voldemort and the folk-tale version of Satan, use your greed to trick you into acting against your own values, then grab away the promised reward at the last moment. But even demons and dark wizards can only do this once or twice before most victims wise up and decide that taking their advice is a bad idea. Game theory can force you to betray your deepest principles for no lasting benefit again and again, and still leave you convinced that your behavior was rational.
Some of the examples in this post probably wouldn’t work in reality; they’re more of a reductio ad absurdum of the so-called homo economicus who acts free from any feelings of altruism or trust. But others are lifted directly from real life where seemingly intelligent people genuinely fall for them. And even the ones that don’t work with real people might be valuable in modeling institutions or governments.
Of the following examples, the first three are from The Art of Strategy; the second three are relatively classic problems taken from around the Internet. A few have been mentioned in the comments here already and are reposted for people who didn’t catch them the first time.
The Evil Plutocrat
You are an evil plutocrat who wants to get your pet bill—let’s say a law that makes evil plutocrats tax-exempt—through the US Congress. Your usual strategy would be to bribe the Congressmen involved, but that would be pretty costly—Congressmen no longer come cheap. Assume all Congressmen act in their own financial self-interest, but that absent any financial self-interest they will grudgingly default to honestly representing their constituents, who hate your bill (and you personally). Is there any way to ensure Congress passes your bill, without spending any money on bribes at all?
Yes. Simply tell all Congressmen that if your bill fails, you will donate some stupendous amount of money to whichever party gave the greatest percent of their votes in favor.
Suppose the Democrats try to coordinate among themselves. They say “If we all oppose the bill, then if even one Republican supports the bill, the Republicans will get lots of money they can spend on campaigning against us. If only one of us supports the bill, the Republicans may anticipate this strategy and two of them may support it. The only way to ensure the Republicans don’t gain a massive windfall and wipe the floor with us next election is for most of us to vote for the bill.”
Meanwhile, in their meeting, the Republicans think the same thing. The vote ends with most members of Congress supporting your bill, and you don’t end up having to pay any money at all.
The Hostile Takeover
You are a ruthless businessman who wants to take over a competitor. The competitor’s stock costs $100 a share, and there are 1000 shares, distributed among a hundred investors who each own ten. That means the company ought to cost $100,000, but you don’t have $100,000. You only have $98,000. Worse, another competitor with $101,000 has made an offer for greater than the value of the company: they will pay $101 per share if they end up getting all of the shares. Can you still manage to take over the company?
Yes. You can make what is called a two-tiered offer. Suppose all investors get a chance to sell shares simultaneously. You will pay $105 for 500 shares—better than they could get from your competitor—but only pay $90 for the other 500. If you get fewer than 500 shares, all will sell for $105; if you get more than 500, you will start by distributing the $105 shares evenly among all investors who sold to you, and then distribute out as many of the $90 shares as necessary (leaving some $90 shares behind except when all investors sell to you) . And you will do this whether or not you succeed in taking over the company—if only one person sells you her share, then that one person gets $105.
Suppose an investor believes you’re not going to succeed in taking over the company. That means you’re not going to get over 50% of shares. That means the offer to buy 500 shares for $105 will still be open. That means the investor can either sell her share to you (for $105) or to your competitor (for $101). Clearly, it’s in this investor’s self-interest to sell to you.
Suppose the investor believes you will succeed in taking over the company. That means your competitor will not take over the company, and its $101 offer will not apply. That means that the new value of the shares will be $90, the offer you’ve made for the second half of shares. So they will get $90 if they don’t sell to you. How much will they get if they do sell to you? They can expect half of their ten shares to go for $105 and half to go for $90; they will get a total of $97.50 per share. $97.50 is better than $90, so their incentive is to sell to you.
Suppose the investor believes you are right on the cusp of taking over the company, and her decision will determine the outcome. In that case, you have at most 499 shares. When the investor gives you her 10 shares, you will end up with 509 − 500 of which are $105 shares and 9 of which are $90 shares. If these are distributed randomly, investors can expect to make on average $104.73 per share, compared to $101 if your competitor buys the company.
Since all investors are thinking along these lines, they all choose to buy shares from you instead of your competitor. You pay out an average of $97.50 per share, and take over the company for $97,500, leaving $500 to spend on the victory party.
The stockholders, meanwhile, are left wondering why they just all sold shares for $97.50 when there was someone else who was promising them $101.
The Hostile Takeover, Part II
Your next target is a small family-owned corporation that has instituted what they consider to be invincible protection against hostile takeovers. All decisions are made by the Board of Directors, who serve for life. Although shareholders vote in the new members of the Board after one of them dies or retires, Board members can hang on for decades. And all decisions about the Board, impeachment of its members, and enforcement of its bylaws are made by the Board itself, with members voting from newest to most senior.
So you go about buying up 51% of the stock in the company, and sure enough, a Board member retires and is replaced by one of your lackeys. This lackey can propose procedural changes to the Board, but they have to be approved by majority vote. And at the moment the other four directors hate you with a vengeance, and anything you propose is likely to be defeated 4-1. You need those other four windbags out of there, and soon, but they’re all young and healthy and unlikely to retire of their own accord.
The obvious next step is to start looking for a good assassin. But if you can’t find one, is there any way you can propose mass forced retirement to the Board and get them to approve it by majority vote? Even better, is there any way you can get them to approve it unanimously, as a big “f#@& you” to whoever made up this stupid system?
Yes. Your lackey proposes as follows: “I move that we vote upon the following: that if this motion passes unanimously, all members of the of the Board resign immediately and are given a reasonable compensation; that if this motion passes 4-1 that the Director who voted against it must retire without compensation, and the four directors who voted in favor may stay on the Board; and that if the motion passes 3-2, then the two ‘no’ voters get no compensation and the three ‘yes’ voters may remain on the board and will also get a spectacular prize—to wit, our company’s 51% share in your company divided up evenly among them.”
Your lackey then votes “yes”. The second newest director uses backward reasoning as follows:
Suppose that the vote were tied 2-2. The most senior director would prefer to vote “yes”, because then she gets to stay on the Board and gets a bunch of free stocks.
But knowing that, the second most senior director (SMSD) will also vote ‘yes’. After all, when the issue reaches the SMSD, there will be one of the following cases:
1. If there is only one yes vote (your lackey’s), the SMSD stands to gain from voting yes, knowing that will produce a 2-2 tie and make the most senior director vote yes to get her spectacular compensation. This means the motion will pass 3-2, and the SMSD will also remain on the board and get spectacular compensation if she votes yes, compared to a best case scenario of remaining on the board if she votes no.
2. If there are two yes votes, the SMSD must vote yes—otherwise, it will go 2-2 to the most senior director, who will vote yes, the motion will pass 3-2, and the SMSD will be forced to retire without compensation.
3. And if there are three yes votes, then the motion has already passed, and in all cases where the second most senior director votes “no”, she is forced to retire without compensation. Therefore, the second most senior director will always vote “yes”.
Since your lackey, the most senior director, and the second most senior director will always vote “yes”, we can see that the other two directors, knowing the motion will pass, must vote “yes” as well in order to get any compensation at all. Therefore, the motion passes unanimously and you take over the company at minimal cost.
The Dollar Auction
You are an economics professor who forgot to go to the ATM before leaving for work, and who has only $20 in your pocket. You have a lunch meeting at a very expensive French restaurant, but you’re stuck teaching classes until lunchtime and have no way to get money. Can you trick your students into giving you enough money for lunch in exchange for your $20, without lying to them in any way?
Yes. You can use what’s called an all-pay auction, in which several people bid for an item, as in a traditional auction, but everyone pays their bid regardless of whether they win or lose (in a common variant, only the top two bidders pay their bids).
Suppose one student, Alice, bids $1. This seems reasonable—paying $1 to win $20 is a pretty good deal. A second student, Bob, bids $2. Still a good deal if you can get a twenty for a tenth that amount.
The bidding keeps going higher, spurred on by the knowledge that getting a $20 for a bid of less than $20 would be pretty cool. At some point, maybe Alice has bid $18 and Bob has bid $19.
Alice thinks: “What if I raise my bid to $20? Then certainly I would win, since Bob would not pay more than $20 to get $20, but I would only break even. However, breaking even is better than what I’m doing now, since if I stay where I am Bob wins the auction and I pay $18 without getting anything.” Therefore Alice bids $20.
Bob thinks “Well, it sounds pretty silly to bid $21 for a twenty dollar bill. But if I do that and win, I only lose a dollar, as opposed to bowing out now and losing my $19 bid.” So Bob bids $21.
Alice thinks “If I give up now, I’ll lose a whole dollar. I know it seems stupid to keep going, but surely Bob has the same intuition and he’ll give up soon. So I’ll bid $22 and just lose two dollars...”
It’s easy to see that the bidding could in theory go up with no limits but the players’ funds, but in practice it rarely goes above $200.
...yes, $200. Economist Max Bazerman claims that of about 180 such auctions, seven have made him more than $100 (ie $50 from both players) and his highest take was $407 (ie over $200 from both players).
In any case, you’re probably set for lunch. If you’re not, take another $20 from your earnings and try again until you are—the auction gains even more money from people who have seen it before than it does from naive bidders (!) Bazerman, for his part, says he’s made a total of $17,000 from the exercise.
At that point you’re starting to wonder why no one has tried to build a corporation around this, and unsurprisingly, the online auction site Swoopo appears to be exactly that. More surprisingly, they seem to have gone bankrupt last year, suggesting that maybe H.L. Mencken was wrong and someone has gone broke underestimating people’s intelligence.
The Bloodthirsty Pirates
You are a pirate captain who has just stolen $17,000, denominated entirely in $20 bills, from a very smug-looking game theorist. By the Pirate Code, you as the captain may choose how the treasure gets distributed among your men. But your first mate, second mate, third mate, and fourth mate all want a share of the treasure, and demand on threat of mutiny the right to approve or reject any distribution you choose.You expect they’ll reject anything too lopsided in your favor, which is too bad, because that was totally what you were planning on.
You remember one fact that might help you—your crew, being bloodthirsty pirates, all hate each other and actively want one another dead. Unfortunately, their greed seems to have overcome their bloodlust for the moment, and as long as there are advantages to coordinating with one another, you won’t be able to turn them against their fellow sailors. Doubly unfortunately, they also actively want you dead.
You think quick. “Aye,” you tell your men with a scowl that could turn blood to ice, “ye can have yer votin’ system, ye scurvy dogs” (you’re that kind of pirate). “But here’s the rules: I propose a distribution. Then you all vote on whether or not to take it. If a majority of you, or even half of you, vote ‘yes’, then that’s how we distribute the treasure. But if you vote ‘no’, then I walk the plank to punish me for my presumption, and the first mate is the new captain. He proposes a new distribution, and again you vote on it, and if you accept then that’s final, and if you reject it he walks the plank and the second mate becomes the new captain. And so on.”
Your four mates agree to this proposal. What distribution should you propose? Will it be enough to ensure your comfortable retirement in Jamaica full of rum and wenches?
Yes. Surprisingly, you can get away with proposing that you get $16,960, your first mate gets nothing, your second mate gets $20, your third mate gets nothing, and your fourth mate gets $20 - and you will still win 3 −2.
The fourth mate uses backward reasoning like so: Suppose there were only two pirates left, me and the third mate. The third mate wouldn’t have to promise me anything, because if he proposed all $17,000 for himself and none for me, the vote would be 1-1 and according to the original rules a tie passes. Therefore this is a better deal than I would get if it were just me and the third mate.
But suppose there were three pirates left, me, the third mate, and the second mate. Then the second mate would be the new captain, and he could propose $16,980 for himself, $0 for the third mate, and $20 for me. If I vote no, then it reduces to the previous case in which I get nothing. Therefore, I should vote yes and get $20. Therefore, the final vote is 2-1 in favor.
But suppose there were four pirates left: me, the third mate, the second mate, and the first mate. Then the first mate would be the new captain, and he could propose $16,980 for himself, $20 for the third mate, $0 for the second mate, and $0 for me. The third mate knows that if he votes no, this reduces to the previous case, in which he gets nothing. Therefore, he should vote yes and get $20. Therefore, the final vote is 2-2, and ties pass.
(He might also propose $16980 for himself, $0 for the second mate, $0 for the third mate, and $20 for me. But since he knows I am a bloodthirsty pirate who all else being equal wants him dead, I would vote no since I could get a similar deal from the third mate and make the first mate walk the plank in the bargain. Therefore, he would offer the $20 to the third mate.)
But in fact there are five pirates left: me, the third mate, the second mate, the first mate, and the captain. The captain has proposed $16,960 for himself, $20 for the second mate, and $20 for me. If I vote no, this reduces to the previous case, in which I get nothing. Therefore, I should vote yes and get $20.
(The captain would avoid giving the $20s to the third and fourth rather than to the second and fourth mates for a similar reason to the one given in the previous example—all else being equal, the pirates would prefer to watch him die.)
The second mate thinks along the same lines and realizes that if he votes no, this reduces to the case with the first mate, in which the second mate also gets nothing. Therefore, he too votes yes.
Since you, as the captain, obviously vote yes as well, the distribution passes 3-2. You end up with $16,980, and your crew, who were so certain of their ability to threaten you into sharing the treasure, each end up with either a single $20 or nothing.
The Prisoners’ Dilemma, Redux
This sequence previously mentioned the popularity of Prisoners’ Dilemmas as gimmicks on TV game shows. In one program, Golden Balls, contestants do various tasks that add money to a central “pot”. By the end of the game, only two contestants are left, and are offered a Prisoners’ Dilemma situation to split the pot between them. If both players choose to “Split”, the pot is divided 50-50. If one player “Splits” and the other player “Steals”, the stealer gets the entire pot. If both players choose to “Steal”, then no one gets anything. The two players are allowed to talk to each other before making a decision, but like all Prisoner’s Dilemmas, the final choice is made simultaneously and in secret.
You are a contestant on this show. You are actually not all that evil—you would prefer to split the pot rather than to steal all of it for yourself—but you certainly don’t want to trust the other guy to have the same preference. In fact, the other guy looks a bit greedy. You would prefer to be able to rely on the other guy’s rational self-interest rather than on his altruism. Is there any tactic you can use before the choice, when you’re allowed to communicate freely, in order to make it rational for him to cooperate?
Yes. In one episode of Golden Balls, a player named Nick successfully meta-games the game by transforming it from the Prisoner’s Dilemma (where defection is rational) to the Ultimatum Game (where cooperation is rational)
Nick tells his opponent: “I am going to choose ‘Steal’ on this round.” (He then immediately pressed his button; although the show hid which button he pressed, he only needed to demonstrate that he had committed and his mind could no longer be changed) “If you also choose ‘Steal’, then for certain neither of us gets any money. If you choose ‘Split’, then I get all the money, but immediately after the game, I will give you half of it. You may not trust me on this, and that’s understandable, but think it through. First, there’s no less reason to think I’m trustworthy than if I had just told you I pressed ‘Split’ to begin with, the way everyone else on this show does. And second, now if there’s any chance whatsoever that I’m trustworthy, then that’s some chance of getting the money—as opposed to the zero chance you have of getting the money if you choose ‘Steal’.”
Nick’s evaluation is correct. His opponent can either press ‘Steal’, with a certainty of getting zero, or press ‘Split’, with a nonzero probability of getting his half of the pot depending on Nick’s trustworthiness.
But this solution is not quite perfect, in that one can imagine Nick’s opponent being very convinced that Nick will cheat him, and deciding he values punishing this defection more than the tiny chance that Nick will play fair. That’s why I was so impressed to see cousin_it propose what I think is an even better solution on the Less Wrong thread on the matter:
This game has multiple Nash equilibria and cheap talk is allowed, so correlated equilibria are possible. Here’s how you implement a correlated equilibrium if your opponent is smart enough:
”We have two minutes to talk, right? I’m going to ask you to flip a coin (visibly to both of us) at the last possible moment, the exact second where we must cease talking. If the coin comes up heads, I promise I’ll cooperate, you can just go ahead and claim the whole prize. If the coin comes up tails, I promise I’ll defect. Please cooperate in this case, because you have nothing to gain by defecting, and anyway the arrangement is fair, isn’t it?”
This sort of clever thinking is, in my opinion, the best that game theory has to offer. It shows that game theory need not be only a tool of evil for classical figures of villainy like bloodthirsty pirate captains or corporate raiders or economists, but can also be used to create trust and ensure cooperation between parties with common interests.