Game theory is applied in a number of fields including business, finance, economics, political science and psychology. Understanding game theory strategies – both the popular ones and some of the relatively lesser-known stratagems – is important to enhance one's reasoning and decision-making skills in a complex world.
Prisoner's Dilemma – In a Nutshell
One of the most popular and basic game theory strategies is Prisoner's Dilemma. This concept explores the decision-making strategy taken by two individuals who, by acting in their own individual best interest, end up with worse outcomes than if they had cooperated with each other in the first place.
In the Prisoner's Dilemma, two suspects who have been apprehended for a crime are held in separate rooms and cannot communicate with each other. The prosecutor informs each of them individually that if he (call him Suspect 1) confesses and testifies against the other, he can go free, but if he does not cooperate and Suspect 2 does, Suspect 1 will be sentenced to three years in prison. If both confess, they will get a two-year sentence, and if neither confesses, they will be sentenced to one year in prison.
While cooperation is the best strategy for the two suspects, when confronted with such a dilemma, research shows that most rational people prefer to confess and testify against the other person rather than stay silent and take the chance that the other party confesses.
Game Theory Strategies
The Prisoner! 's Dilemma lays the foundation for advanced game theory strategies of which the popular ones include:
Matching Pennies: This is a zero-sum game that involves two players (call them Player A and Player B) simultaneously placing a penny on the table, with the payoff depending on whether the pennies match. If both pennies are heads or tails, Player A wins and keeps Player B's penny. If they do not match, Player B wins and keeps Player A's penny. Deadlock: This is a social dilemma scenario like Prisoner's Dilemma in that two players can either cooperate or defect (i.e. not cooperate). In Deadlock, if Player A and Player B both cooperate, they each get a payoff of 1, and if they both defect, they each get a payoff of 2. But if Player A cooperates and Player B defects, then A gets a payoff of 0 and B gets a payoff of 3. In the payoff diagram below, the first numeral in the cells (a) through (d) represents Player A's payoff, and the second numeral is that of Player B:
Deadlock Payoff Matrix |
Player B |
||
Cooperate |
Defect |
||
Player A |
Cooperate |
(a) 1, 1 |
(b) 0, 3 |
Defect |
(c) 3, 0 |
(d) 2, 2 |
A commonly cited example of Deadlock is that of two nuclear powers trying to reach an agreement to eliminate their arsenals of nuclear bombs. In this case, cooperation implies adhering to the agreement, while defection means secretly reneging on the agreement and retaining the nuclear arsenal. The best outcome for either nation, unfortunately, is to renege on the agreement and retain the nuclear option while the other nation eliminates its arsenal, since this will give the former a tremendous hidden advantage over the latter if war ever breaks out between the two. The second-best option is for both to defect or not cooperate, since this retains their status as nuclear powers.
Cournot Competition: This model is also conceptually similar to Prisoner's Dilemma, and is named after French mathematician Augustin Cournot, who introduced it in 1838. The most common application of the Cournot model is in describing a duopoly or two main producers in a market. For example, assume two companies A and B produce an identical product and can produce high or low quantities. If they both cooperate and agree to produce at low levels, then limited supply will translate into a high price for the product on the market and substantial profits for both companies. On the other hand, if they defect and produce at high levels, the market will be swamped and result in a low price for the product and consequently lower profits. But if one cooperates (i.e. produces at low levels) and the other defects (i.e. surreptitiously produces at high levels), then the former just breaks even while the latter earns a profit that is higher than if they both cooperate.
The payoff matrix for companies A and B is shown (figures represent profit in millions of dollars). Thus, if A cooperates and produces at low levels while B defects and produces at high levels, the payoff is as shown in cell (b) – break-even for company A and $7 million in profits for company B.
Cournot Payoff Matrix |
Company B |
||
Cooperate |
Defect |
||
Company A |
Cooperate |
(a) 4, 4 |
(b) 0, 7 |
Defect |
(c) 7, 0 |
(d) 2, 2 |
Thus, if both companies decide to introduce the new technology, they would earn $600 million apiece, while introducing a revised version of the older technology would earn them $300 million each, as shown in cell (d). But if Company A decides alone to introduce the new technology, it would only earn $150 million, even though Company B would earn $0 (presumably because consumers may not be willing to pay for its now-obsolete technology). In this case, it makes sense for both companies to work together rather than on their own.
Coordination Payoff Matrix |
Company B |
||
New technology |
Old technology |
||
Company A |
New technology |
(a) 600, 600 |
(b) 0, 150 |
Old technology |
(c) 150, 0 |
(d) 300, 300 |
The point of the game is that if A and B both cooperate and "pass" to the end of the game, they get the maximum payoff of $100 each. But if they distrust the other player and expect them to "take" at the first opportunity, then the Nash equilibrium predicts that players will take the lowest possible claim ($1 in this case). Experimental studies have shown, however, that this "rational" behavior (as predicted by game theory) is seldom exhibited in real life. This is not intuitively surprising given the tiny size of the initial payoff in relation to the final one. Similar behavior by experimental subjects has also been exhibited in the Traveler's Dilemma.
Traveler's Dilemma: This is a non-zero sum game in which both players attempt to maximize their own payoff without regard to the other. Devised by economist Kaushik Basu in 1994, in Traveler's Dilemma, an airline agrees to pay two travelers compensation for damages to identical items. However, the two travelers are separately required to estimate the value of the item, with a minimum of $2 and a maximum of $100. If both write down the same value, the airline will reimburse each of them that amount. But if the values differ, the airline will pay them the lower value, with a bonus of $2 for the traveler who wrote down this lower value and a penalty of $2 for the traveler who wrote down the higher value. The Nash equilibrium level, based on backward induction, is $2 in this scenario. But as in the Centipede game, laboratory experiments consistently demonstrate that most participants – naively or otherwise – pick a number much higher than $2.
Traveler's Dilemma can be applied to analyze a variety of real-life situations. The process of backward induction, for example, can help explain how two companies engaged in cutthroat competition can steadily ratchet product prices lower in a bid to gain market share, which may result in them incurring increasingly greater losses in the process.
Additional Game Theory Strategies
Battle of the Sexes: This is another form of the coordination game described earlier but with some payoff asymmetries. It essentially involves a couple trying to coordinate their evening out. While they had agreed to meet at either the ball game (the man's preference) or at a play (the woman's preference), they have forgotten what they had decided, and to compound the problem, cannot communicate with one another. Where should they go? The payoff matrix is as shown – the numerals in the cells represent the relative degree of enjoyment of the event for the woman and man, respectively. For example, cell (a) represents the payoff (in terms of enjoyment levels) for the woman and man, respectively, at the play (she enjoys it much more than he does). Cell (d) is the payoff if both make it to the ball game (he enjoys it more than she does). Cell (c) represents the dissatisfaction if both go not only to the wrong location, but also to the event they enjoy least – the woman to the ball game and the man to the play.
Battle of the Sexes Payoff Matrix |
Man |
||
Play |
Ball game |
||
Woman |
Play |
(a) 6,3 |
(b) 2, 2 |
Ball game |
(c) 0, 0 |
(d) 3, 6 |
Peace-war Payoff Matrix |
Company B |
||
Peace |
War |
||
Company A |
Peace |
(a) 3,3 |
(b) 0,4 |
War |
(c) 4,0 |
(d) 1,1 |
Game theory can be used very effectively as a tool for decision-making whether in an economical, business or personal setting.
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