To employ a grim trigger strategy effectively, it is important that the player communicate their intentions before the first defection takes place. Once a defection occurs, cooperation cannot be restored. 'Grim Trigger' is essentially the nuclear weapon of strategies--it threatens mutual destruction to ensure cooperation and is therefore a more effective threat than tactic. To successfully force cooperation using a 'grim trigger' strategy is it important that opponents believe in the credibility of your commitment to the strategy--whether its true or not.
Showing posts with label game theory. Show all posts
Showing posts with label game theory. Show all posts
Wednesday, February 10, 2010
Grim Trigger
Grim trigger is a strategy in game theory for a repeated game, such as an iterated prisoner's dilemma. Initially, a player using grim trigger will cooperate, but as soon as the opponent defects (thus satisfying the trigger condition), the player using grim trigger will defect for the remainder of the iterated game. Since a single defect by the opponent triggers defection forever, grim trigger is the most strictly unforgiving of strategies in an iterated game.
Monday, August 17, 2009
Chicken
The game of Chicken models two drivers, both headed for a single lane bridge from opposite directions. The first to swerve away yields the bridge to the other. If neither player swerves, the result is a costly deadlock in the middle of the bridge, or a potentially fatal head-on collision. It is presumed that the best thing for each driver is to stay straight while the other swerves (since the other is the "chicken" while a crash is avoided). Additionally, a crash is presumed to be the worst outcome for both players. This yields a situation where each player, in attempting to secure his best outcome, risks the worst.
Because the loss of swerving is so trivial compared to the crash that occurs if nobody swerves, the reasonable strategy would seem to be to swerve before a crash is likely. Yet, knowing this, if one believes one's opponent to be reasonable, one may well decide not to swerve at all, in the belief that he will be reasonable and decide to swerve, leaving the other player the winner. The game is similar to the prisoner's dilemma game in that an "agreeable" mutual solution is unstable since both players are individually tempted to stray from it.
One tactic in the game is for one party to signal their intentions convincingly before the game begins. For example, if one party were to remove their steering wheel just before the match, the other party would be compelled to swerve. This shows that, in some circumstances, reducing one's own options can be a good strategy. One real-world example is a protester who handcuffs himself to an object, so that no threat can be made which would compel him to move (since he cannot move).
Because the loss of swerving is so trivial compared to the crash that occurs if nobody swerves, the reasonable strategy would seem to be to swerve before a crash is likely. Yet, knowing this, if one believes one's opponent to be reasonable, one may well decide not to swerve at all, in the belief that he will be reasonable and decide to swerve, leaving the other player the winner. The game is similar to the prisoner's dilemma game in that an "agreeable" mutual solution is unstable since both players are individually tempted to stray from it.
One tactic in the game is for one party to signal their intentions convincingly before the game begins. For example, if one party were to remove their steering wheel just before the match, the other party would be compelled to swerve. This shows that, in some circumstances, reducing one's own options can be a good strategy. One real-world example is a protester who handcuffs himself to an object, so that no threat can be made which would compel him to move (since he cannot move).
Thursday, June 18, 2009
Invisible Hand
The invisible hand is a metaphor coined by the economist Adam Smith in The Wealth of Nations. In economics, the invisible hand is the term economists use to describe the self-regulating nature of the marketplace. More broadly, it describes any situation where multiple people cooperate solely out of self interest rather than any altruistic motive.
Smith provides an example that illustrates the simplicity of the principle: “It is not from the benevolence of the butcher, the brewer or the baker, that we expect our dinner, but from their regard to their own self interest. We address ourselves, not to their humanity but to their self-love, and never talk to them of our own necessities but of their advantages."
This is an example of a game theory situation known as a "stag hunt." The stag hunt is a close cousin to the Prisoner's Dilemma. Jean-Jacques Rousseau described a situation in which two individuals go out on a hunt. Each can individually choose to hunt a stag or hunt a hare. Each player must choose an action without knowing the choice of the other. If an individual hunts a stag, he must have the cooperation of his partner in order to succeed. An individual can get a hare by himself, but a hare is worth less than a stag.
In addition to the example suggested by Rousseau, David Hume provides an example of a stag hunts. His addresses two individuals who must row a boat. If both choose to row they can successfully move the boat. However if one doesn't, the other wastes his effort.
Smith provides an example that illustrates the simplicity of the principle: “It is not from the benevolence of the butcher, the brewer or the baker, that we expect our dinner, but from their regard to their own self interest. We address ourselves, not to their humanity but to their self-love, and never talk to them of our own necessities but of their advantages."
This is an example of a game theory situation known as a "stag hunt." The stag hunt is a close cousin to the Prisoner's Dilemma. Jean-Jacques Rousseau described a situation in which two individuals go out on a hunt. Each can individually choose to hunt a stag or hunt a hare. Each player must choose an action without knowing the choice of the other. If an individual hunts a stag, he must have the cooperation of his partner in order to succeed. An individual can get a hare by himself, but a hare is worth less than a stag.
In addition to the example suggested by Rousseau, David Hume provides an example of a stag hunts. His addresses two individuals who must row a boat. If both choose to row they can successfully move the boat. However if one doesn't, the other wastes his effort.
Thursday, May 28, 2009
God's Algorithm
God's algorithm is a way to solve puzzles and games using the least possible number of moves, the idea being that an omniscient being would know an optimal step from any given configuration. God's algorithm is essentially the most efficient strategy for any given game, one that cannot be improved upon in any way. The solutions to the recently posted river crossing puzzles are an example.
Tic-tac-toe and Tower of Hanoi are all examples of games with simple God's algorithms. Most children figure out the ideal strategy themselves pretty quickly. One of the reasons children like these games is once they figure out the optimal strategy, no one has an edge on them--not adults, not their parents, not their teachers. They are 'experts' at the game. In fact, someone who knows the God's algorithm for a game could best God himself in a fair match. Even the eyes of God see no more to a game of tic-tac-toe than a capable player. This has lead to a wealth of fiction where supernatural beings are bested by humans in games of strategy and chance.
God's algorithms have been suggested for Rubick's cubes, chess, Irensei and Go. Theoretically any game with perfect information should have a God's algorithim solution. A game is said to have perfect information if all players know all moves that have taken place. For instance, on a chessboard there are no secrets. You and your opponent can see all of the pieces on the board at all times. By contrast, poker is a game with imperfect information since you can neither see your opponent's cards nor know which cards will come out the deck next.
Tic-tac-toe and Tower of Hanoi are all examples of games with simple God's algorithms. Most children figure out the ideal strategy themselves pretty quickly. One of the reasons children like these games is once they figure out the optimal strategy, no one has an edge on them--not adults, not their parents, not their teachers. They are 'experts' at the game. In fact, someone who knows the God's algorithm for a game could best God himself in a fair match. Even the eyes of God see no more to a game of tic-tac-toe than a capable player. This has lead to a wealth of fiction where supernatural beings are bested by humans in games of strategy and chance.
God's algorithms have been suggested for Rubick's cubes, chess, Irensei and Go. Theoretically any game with perfect information should have a God's algorithim solution. A game is said to have perfect information if all players know all moves that have taken place. For instance, on a chessboard there are no secrets. You and your opponent can see all of the pieces on the board at all times. By contrast, poker is a game with imperfect information since you can neither see your opponent's cards nor know which cards will come out the deck next.
Wednesday, May 13, 2009
The Peter Principle
The Peter Principle says, "In a Hierarchy Every Employee Tends to Rise to His Level of Incompetence." Members are promoted so long as they work competently; Sooner or later they are promoted to a position at which they are no longer competent (their "level of incompetence"), and there they remain, being unable to earn further promotions. Peter's Corollary states that "in time, every post tends to be occupied by an employee who is incompetent to carry out his duties" and adds that "work is accomplished by those employees who have not yet reached their level of incompetence".
An example:
If you're a proficient and effective accountant, you're most likely demonstrating peak competence in your job right now. As a result of your performance, your valuable contribution results in a promotion to a management position. In this new position, you now do few of the original tasks which gained you acclaim. Given this, promotions stop, and there you stay, until you retire.
A dramatic example of the Peter Principle at work is Michael Scott from the Office. Michael is frequently shown to be an impressive salesman but an utterly inept manager. He has been promoted to his level of incompetence, where he will now remain.Monday, May 11, 2009
Going to the Movies
Here's a situation we've all seen before; suppose Alice and Bob have to decide whether to go to the movies to see a chick flick, and that each has the liberty to decide whether to go themselves. If the personal preferences are based on Alice first wanting to be with Bob, then thinking it is a good film, and on Bob first wanting Alice to see it but then not wanting to go himself, then the personal preference orders might be:
How will these preferences play out? Bob will not go on his own: he would not set off alone, but if for some reason he did, then Alice would follow because she prefers both to go > Bob to go. Alternatively, if Alice decided to go alone, Bob would not join because that is his most desired outcome because he prefers Alice to go > both to go. However if Bob chooses not to go, Alice will want to stay home too because she prefers neither to go > Alice to go.
Herein lies the rub: even though Alice and Bob prefer both to go > neither to go, if Alice and Bob choose individually, neither will end up going because neither prefers going alone to both staying home. Bob might try to convince Alice to go, since both scenarios he prefers over neither going have Alice go; but Alice can't convince Bob to go because as soon she's going Bob can achieve his optimal outcome by staying home.
- Alice wants: both to go > neither to go > Alice to go > Bob to go
- Bob wants: Alice to go > both to go > neither to go > Bob to go
How will these preferences play out? Bob will not go on his own: he would not set off alone, but if for some reason he did, then Alice would follow because she prefers both to go > Bob to go. Alternatively, if Alice decided to go alone, Bob would not join because that is his most desired outcome because he prefers Alice to go > both to go. However if Bob chooses not to go, Alice will want to stay home too because she prefers neither to go > Alice to go.
Herein lies the rub: even though Alice and Bob prefer both to go > neither to go, if Alice and Bob choose individually, neither will end up going because neither prefers going alone to both staying home. Bob might try to convince Alice to go, since both scenarios he prefers over neither going have Alice go; but Alice can't convince Bob to go because as soon she's going Bob can achieve his optimal outcome by staying home.
- Probable outcomes: neither go > Alice goes > both go > Bob goes.
Friday, May 8, 2009
Code of Silence: Two Perspectives
Whether it's the Mafia or the Underground Railroad, codes of silence are vital to some organizations' continued livelihood. This post focuses on one aspect of the code of silence: no snitching. Are tight knit groups of individuals who trust one another in turn able to work more productively or is the code of silence just a clever ploy to get kingpins off the hook while their subordinates take the fall?
Take One: Manipulated Minions
Codes of silence are philosophically inconsistent. The implications of a code of silence are all about putting the group over the individual, but at the same time they limit the group so narrowly that they do not serve the greater good in any general sense.
If you believe in a way of conduct, you shouldn't have to conceal it--whenever people rely on secrecy, it's a warning they might be up to no good. Granted that the underground railroad might be an exception to this rule, but the underground railroad was underpinned by a broader position: that slavery was wrong. Criminal organizations however don't have that kind of broad message underneath: they're not trying to encourage everyone to become a criminal like the underground railroad was trying to encourage everyone to become an abolitionist.
For criminal organizations, the only underlying interest is personal gain--which is why a code of silence is inconsistent because its advocating the good of the group over protecting the self. The only way it can be explained is if you can narrowly tailor it to mean 'protection of this groups interests are more important than the individual but less important than everyone else's, but since the group in question's interest is self interest, it doesn't really add up as a cohesive position.
Codes of silence are only good for the kingpin, not the individual and certainly not society's. Kingpins promulgate the code because it favors them, and often artificially increase the incentives to cooperate by threatening to retaliate against people who break it. Silence is not a very philosophically sound system which is why it cant be sustained without killing people or otherwise redistributing the costs. From a game theory standpoint, its a preferable position to advocate if you RUN a criminal organization because it protects your self interest while masking itself as a broader principle.
Take Two: Real Life Prisoner's Dilemma
For anybody on earth not familiar with the prisoner's dilemma, here's a little recap:
Two suspects are arrested by the police. The police have insufficient evidence for a conviction, and, having separated both prisoners, visit each of them to offer the same deal. If one testifies (defects from the other) for the prosecution against the other and the other remains silent (cooperates with the other), the betrayer goes free and the silent accomplice receives the full 10-year sentence. If both remain silent, both prisoners are sentenced to only six months in jail for a minor charge. If each betrays the other, each receives a five-year sentence. Each prisoner must choose to betray the other or to remain silent. Each one is assured that the other would not know about the betrayal before the end of the investigation. How should the prisoners act?
What's remarkable about the prisoner's dilemma is the way it exploits our self interest. No matter what the other player does, one player will always gain a greater payoff by defecting. Since in any situation playing defect is more beneficial than cooperating, all rational players will play defect, all things being equal--even though we know the sentence would be only 1/10 as long if they cooperated.
A code of silence is one possible solution to the the prisoner's dilemma. Kingpins may be able to see the best outcome because they have no personal stake in the pot (experience helps too, a good way to induce cooperation is have people play the game over and over again).
Take One: Manipulated Minions
Codes of silence are philosophically inconsistent. The implications of a code of silence are all about putting the group over the individual, but at the same time they limit the group so narrowly that they do not serve the greater good in any general sense.
If you believe in a way of conduct, you shouldn't have to conceal it--whenever people rely on secrecy, it's a warning they might be up to no good. Granted that the underground railroad might be an exception to this rule, but the underground railroad was underpinned by a broader position: that slavery was wrong. Criminal organizations however don't have that kind of broad message underneath: they're not trying to encourage everyone to become a criminal like the underground railroad was trying to encourage everyone to become an abolitionist.
For criminal organizations, the only underlying interest is personal gain--which is why a code of silence is inconsistent because its advocating the good of the group over protecting the self. The only way it can be explained is if you can narrowly tailor it to mean 'protection of this groups interests are more important than the individual but less important than everyone else's, but since the group in question's interest is self interest, it doesn't really add up as a cohesive position.
Codes of silence are only good for the kingpin, not the individual and certainly not society's. Kingpins promulgate the code because it favors them, and often artificially increase the incentives to cooperate by threatening to retaliate against people who break it. Silence is not a very philosophically sound system which is why it cant be sustained without killing people or otherwise redistributing the costs. From a game theory standpoint, its a preferable position to advocate if you RUN a criminal organization because it protects your self interest while masking itself as a broader principle.
Take Two: Real Life Prisoner's Dilemma
For anybody on earth not familiar with the prisoner's dilemma, here's a little recap:
Two suspects are arrested by the police. The police have insufficient evidence for a conviction, and, having separated both prisoners, visit each of them to offer the same deal. If one testifies (defects from the other) for the prosecution against the other and the other remains silent (cooperates with the other), the betrayer goes free and the silent accomplice receives the full 10-year sentence. If both remain silent, both prisoners are sentenced to only six months in jail for a minor charge. If each betrays the other, each receives a five-year sentence. Each prisoner must choose to betray the other or to remain silent. Each one is assured that the other would not know about the betrayal before the end of the investigation. How should the prisoners act?
| OUTCOMES | Prisoner B Stays Silent | Prisoner B Betrays |
|---|---|---|
| Prisoner A Stays Silent | Each serves 6 months | Prisoner A: 10 years Prisoner B: goes free |
| Prisoner A Betrays | Prisoner A: goes free Prisoner B: 10 years | Each serves 3 years |
What's remarkable about the prisoner's dilemma is the way it exploits our self interest. No matter what the other player does, one player will always gain a greater payoff by defecting. Since in any situation playing defect is more beneficial than cooperating, all rational players will play defect, all things being equal--even though we know the sentence would be only 1/10 as long if they cooperated.
A code of silence is one possible solution to the the prisoner's dilemma. Kingpins may be able to see the best outcome because they have no personal stake in the pot (experience helps too, a good way to induce cooperation is have people play the game over and over again).
Saturday, May 2, 2009
The Dollar Auction: Irrational Escalation of Commitment
The dollar auction is a game designed by economist Martin Shubik to illustrate a paradox brought about by traditional rational choice theory in which players with perfect information are compelled to make irrational decisions.
The setup involves an auction for a one dollar bill with the following rule: the dollar goes to the highest bidder, who pays the amount he bids. The second-highest bidder also must pay the highest amount that he bid, but gets nothing in return. The second highest bidder might not have to pay on eBay, but in many real contests, both sides end up paying but only one gets the prize--like lawsuits, sports competitions, gambling and political campaigns.
Bidding when the price is below fifty cents or so seems harmless because it’s an obvious deal to buy a dollar for any amount less. The twist becomes clear about when the high bid is 80 cents. People start to think about how the second rule–the one requiring the loser to pay–would affect incentives. What might the second highest bidder think at this stage? He is offering 70 cents but being outbid. There are two choices he could make:
- do nothing and lose 70 cents if the auction ends
- bid up to 90 cents, and if the auction ends, win the dollar, and profit 10 cents
But this action has an effect on the person bidding 80 cent, who is now the second highest bidder. This person will now make a similar calculation. He can either do nothing and lose 80 cents if the auction ends, or he can raise the bid to a dollar and have a chance of breaking even. Again, bidding higher makes sense. Thinking more generally, it always make sense for the second highest bidder to increase the bid.
Soon people will bid more than one dollar and fight over who will lose less money. It is the incentives that dictate this weird outcome. Consider an example when the highest bid is $1.50. Since the high bid is above the prize of $1, it is clear no new bidder will enter. Hence, the second bidder faces the two choices of doing nothing and losing $1.40, or raising the bid to $1.60 to lose only 60 cents if the auction ends.
In this case, it makes just as much sense to limit loss as it does to seek profit. The second highest bidder will raise the bid. In turn, the other bidder will perform a similar calculation and again raise the top bid. This bidding war can theoretically continue indefinitely. In practical situations, it ends when someone chooses to fold. This game is played at Stanford in economics classes, and its not uncommon to see the game end anywhere between five and ten dollars.
Here are some other real life examples of the irrational escalation of commitment:
- After a heated and aggressive bidding war, Robert Campeau ended up buying Bloomingdale's for an estimated 600 million dollars more than it was worth. The Wall Street Journal noted that "we're not dealing in price anymore but egos." Campeau was forced to declare bankruptcy soon afterwards.
- Supporters of the Iraq War have used the casualties of the conflict in Iraq since 2003 to justify years of further military commitment. This rationale was also used during the sixteen-year Vietnam War, another military example of the logical fallacy.
- Two competing brands often end up spending money on advertising wars without either increasing market share in a significant manner. Though the most commonly cited examples of this are Maxwell House and Folgers in the early 1990s, this has also been seen between Coke and Pepsi, and Kodak and Polaroid.
- Shakespeare's Macbeth comments, "I am in blood stepped in so far that, should I wade no more, returning were as tedious as go o'er." The metaphor represents Macbeth's crimes and rather than stop committing crimes (presumably, for fear of damnation) Macbeth says that he has "passed the point of no return" and might as well continue, even though it will inevitably lead to his downfall.
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