Recently I had the pleasure of applying Game Theory on one of the problems I worked with. Initially I was skeptical about the validity of the theory and felt that it is trying to give an explanation of an already occurring social event (how individuals react to the situation). Later realized the beauty of the theory on why people adopt a specific strategy on a given situation. After all we humans want to maximize our profit or at least get as much as what the opponent will get in the game of life. In "The Beautiful Mind" movie, when the beautiful blonde girl along with her 4 friends comes to the party, the best strategy of John Nash’s 4 friends is to better propose the 4 of her friends rather than the blonde girl. There is a higher probability of the blonde girl denying company to all 4 of them as well as the 4 of her friends being pissed off by the fact they all 4 approached the blonde girl in their covey. [In a lighter vein, of course once 4 of Nash’s friends have given the company to 4 of her friends, Nash has always the gain of giving company to the blonde girl]
The essence of game theory is captured by the famous Prisoner’s Dilemma (PD) experiment. Two suspects (A and B) committed a crime and were arrested. Interrogated separately: they may confess (C) or deny (D). These are their “strategies.” Payoffs are what they get by adopting a particular strategy. By adopting a strategy of D they have the freedom of not going to jail. On the other hand by adopting a strategy of C, at least one of them can escape prison and can also be rewarded for turning approver. “Best” outcome for the PD experiment is for both of them to deny: (D, D) – Pareto Optimal (i.e., the sum of payoffs is maximized). Pareto optimality implies that it is not possible to make somebody better off without making somebody else worse off. This is the whole idea for countries spending on defense and amassing weapons. This creates a sense of fear in the other country to wage a war. [I used to wonder as a kid why countries spend the maximum % of GDP on defense. Now I know why partially. See my conclusion on my personal dilemma]. The equilibrium outcome of the game need not be the Pareto optimal outcome. In the PD there’s always an incentive for a player to deviate from (D, D). In fact, choosing C always gets you a better payoff than choosing D (the black sheep or the one who turns approver escapes from prison sentence). We say that the players have a dominant strategy. Clearly, they will use this strategy in equilibrium. Nash Equilibrium (NE): by definition, a NE is a pair of strategies (SA, SB), so that if A sticks to his NE strategy SA, B has no profitable deviation (i.e., B will do no better if he uses any other strategy). The same holds for A: if B plays his NE strategy, A cannot do better than playing his equilibrium strategy. The NE for PD is (C, C). [There are no unilateral profitable deviations from the Nash equilibrium exist.] Suppose that the game in hand is the twice-repeated PD. How does a strategy look like? What about the strategy in the ten times repeated PD? In the finitely repeated PD game, consider the last period. What would you do if you were A? Go to the next-to-last period. What would you do if you were A? Conclusion: There is no collusion in the finite-period repeated PD. Equilibrium strategies are history-independent here.
What if the game is repeated an infinite (or random) number of times? We will show that if players put enough weight on future streams of income, collusion may be enforced. In PD both players would be better off if they played in each period (D, D). They choose to play C because of the opponent’s incentive to “cheat” by playing C. Mr. A offers the following “plan” to Mr. B: he will start by playing D in the first round; will continue to play D as long as Mr. B plays D; if a deviation is detected, Mr. A will play C forever. This is a “grim” trigger strategy (any deviation from D triggers “punishment” of the opponent – forever!). This won't occur in reality since as humans we are always suspicious of opponent's commitment. This partially explains why Walmart, Kroger have different pricing strategies and fortunately there is no collusion in their pricing strategies. There is always an incentive to collude and set a fixed price. But this cannot last long for an infinitely repeated game because there is always an incentive for say Walmart to lower the price and attract more customers. So it is better off for both the sellers to not collude. Thus we shoppers are protected.
What about mixed strategy in the game of chance like poker or the simple coin toss experiment? These are zero sum games: One person’s gain is another person’s loss. The only way to win in this game is to randomize our strategies: choose H and T with equal probability with no bias.
Having explained the Game Theory, I always felt it has some short comings particularly in explaining one of the critical event in Indian history (my personal dilemma). The strategy adopted by Mahatma Gandhi in freedom struggle. His strategy of adopting non-violence pitted against violence, proved to a dominant strategy. By applying Game Theory, the strategy against violence is violence of greater degree but by adopting a strategy of non-violence he was able to achieve what India badly needed - freedom.
Anyway, start analyzing how you reacted to situations with Game Theory in mind (postmortem analysis). You will be surprised to find your dominant strategy in the given situation.
The essence of game theory is captured by the famous Prisoner’s Dilemma (PD) experiment. Two suspects (A and B) committed a crime and were arrested. Interrogated separately: they may confess (C) or deny (D). These are their “strategies.” Payoffs are what they get by adopting a particular strategy. By adopting a strategy of D they have the freedom of not going to jail. On the other hand by adopting a strategy of C, at least one of them can escape prison and can also be rewarded for turning approver. “Best” outcome for the PD experiment is for both of them to deny: (D, D) – Pareto Optimal (i.e., the sum of payoffs is maximized). Pareto optimality implies that it is not possible to make somebody better off without making somebody else worse off. This is the whole idea for countries spending on defense and amassing weapons. This creates a sense of fear in the other country to wage a war. [I used to wonder as a kid why countries spend the maximum % of GDP on defense. Now I know why partially. See my conclusion on my personal dilemma]. The equilibrium outcome of the game need not be the Pareto optimal outcome. In the PD there’s always an incentive for a player to deviate from (D, D). In fact, choosing C always gets you a better payoff than choosing D (the black sheep or the one who turns approver escapes from prison sentence). We say that the players have a dominant strategy. Clearly, they will use this strategy in equilibrium. Nash Equilibrium (NE): by definition, a NE is a pair of strategies (SA, SB), so that if A sticks to his NE strategy SA, B has no profitable deviation (i.e., B will do no better if he uses any other strategy). The same holds for A: if B plays his NE strategy, A cannot do better than playing his equilibrium strategy. The NE for PD is (C, C). [There are no unilateral profitable deviations from the Nash equilibrium exist.] Suppose that the game in hand is the twice-repeated PD. How does a strategy look like? What about the strategy in the ten times repeated PD? In the finitely repeated PD game, consider the last period. What would you do if you were A? Go to the next-to-last period. What would you do if you were A? Conclusion: There is no collusion in the finite-period repeated PD. Equilibrium strategies are history-independent here.
What if the game is repeated an infinite (or random) number of times? We will show that if players put enough weight on future streams of income, collusion may be enforced. In PD both players would be better off if they played in each period (D, D). They choose to play C because of the opponent’s incentive to “cheat” by playing C. Mr. A offers the following “plan” to Mr. B: he will start by playing D in the first round; will continue to play D as long as Mr. B plays D; if a deviation is detected, Mr. A will play C forever. This is a “grim” trigger strategy (any deviation from D triggers “punishment” of the opponent – forever!). This won't occur in reality since as humans we are always suspicious of opponent's commitment. This partially explains why Walmart, Kroger have different pricing strategies and fortunately there is no collusion in their pricing strategies. There is always an incentive to collude and set a fixed price. But this cannot last long for an infinitely repeated game because there is always an incentive for say Walmart to lower the price and attract more customers. So it is better off for both the sellers to not collude. Thus we shoppers are protected.
What about mixed strategy in the game of chance like poker or the simple coin toss experiment? These are zero sum games: One person’s gain is another person’s loss. The only way to win in this game is to randomize our strategies: choose H and T with equal probability with no bias.
Having explained the Game Theory, I always felt it has some short comings particularly in explaining one of the critical event in Indian history (my personal dilemma). The strategy adopted by Mahatma Gandhi in freedom struggle. His strategy of adopting non-violence pitted against violence, proved to a dominant strategy. By applying Game Theory, the strategy against violence is violence of greater degree but by adopting a strategy of non-violence he was able to achieve what India badly needed - freedom.
Anyway, start analyzing how you reacted to situations with Game Theory in mind (postmortem analysis). You will be surprised to find your dominant strategy in the given situation.
