The Traveler’s Dilemma describes a situation which is seen as somewhat of a paradox by game theorists and economist. Loving a good supposed paradox, I became interested in it after reading a cursory mention in a footnote. After reading the problem, I certainly realized the troubling gap between the game theoretic solution and the common-sense solution. I must examine the literature further on this to see some proposed fixes. This blog isn’t about that though. Often times, when translating technical mathematical statements into easily digestible colloquial speech, the actual mathematical result is oversimplified and loses its meaning. I think a bit of that is occurring here. Let’s look at the dilemma.
Two passengers are returning to the United States from abroad. Upon departure of the plane, the passengers realize that incompetent TSA workers, after invading their privacy and searching their bags, had caused the destruction of an antique had purchased outside the country. The two antiques were identical. The passengers file a claim form with the government seeking compensation. Now the TSA’s claims department, being too lazy to do proper research, does not know the true value of the antique. However, they have a plan. They tell each passenger to write down a value on their claim form between $2 and $100. If both passengers write down the same number x, they will each be rewarded with $x. If two different numbers are claimed, call them m and n (assume m < style="">, the lower value m will be taken as the true value. Being social engineers as the government is one to do, it is decided that the "honest" passenger claiming m, will receive $(m+2) as a reward. The "lying" traveler will receive only $(m-2), with $2 taken away as a punishment. Now, what value should the passenger claim?
Upon my first reading, I naively answered “however much the souvenir cost”, completely missing the paradox. Ditching any sense of morality then, how much should we claim thinking only of our gains? Did you answer $100 dollars? Or a value close to it? You naïve layman with your feeble reasoning skills. The game theoretic solution tells us that each passenger should claim $2, which we denote by (2,2).
The strategy (2,2) is what’s called a Nash Equilibrium. In this sense, we call it a solution to the problem. The paradox manifests itself here since we would intuitively be able to reach a much higher gain by simply claiming a higher price. Yet game theory tells us (2,2) is optimal. A Nash Equilibrium has the property that neither party can improve his gain by unilaterally changing his strategy. To see this assume the first traveler claimed $2 and the second claimed any value other than $2, so that we have (2, y>2). Then, by the game parameters, $2 will be the accepted value. Traveler 1 will gain the $2 value, plus his $2 truth bonus for $4. Traveler 2 will receive $2 minus his $2 penalty for no gain. Similarly, the strategies (x>2, 2) would yield unfortunate results for Traveler 1.
Part of the communication breakdown comes from the meaning of optimal in the vernacular and as a technical definition. In regular speech, an optimal solution would mean the best possible solution. In Mathematics, what we mean is that the strategy will maximize a player’s minimum gain (or equivalently minimize the maximum loss). Claiming $2 clearly does this, as it guarantees a gain of at least $2. Claiming any other price leaves open the scenario where the other player claims $2 and you gain nothing. Keeping this in mind, the solution becomes much less puzzling. It is only a solution in the sense that, assuming some conditions about known information and rationality, it meets a narrow technical definition of what properties we desire in a solution.
Nash equilibriums are helpful when there’s a threat of your opponent outplaying you. If a Nash Equilibrium exists in a given game, you equalize the playing field by employing one. For no matter how outmatched you are by an opponent, you force him into also playing the Nash Equilibrium, essentially removing the skill disparity. In this situation, a Nash Equilibrium may not be the strategy we wish to employ.
I’ll close with one final thought. When we speak of optimal or best solutions in regular life, the closest game theoretic analogue (which I know of at least) is the dominated strategy. We say Strat A dominates Strat B, when the payoff from Strat A is greater than or equal to the payoff for Strat B, regardless of what strategy the opposing player chooses. To give an example: Suppose one is playing Texas Hold’em. On the river you hold a Royal Flush, and your opponent bets into you. You then have three strategy options: Fold, Call, or Raise. The strategy of raising dominates all other strategies. For no matter what cards your opponent holds and no matter how your opponent responds, you will gain more from raising than you do by either calling or folding. Domination is a strong criterion than Nash Equilibrium and rarely applies in real games. Notice that in the Traveler’s Dilemma , no strategy dominates any other. As an example, claiming $2 does not dominate claiming $100 dollars, since the payoff for (2,100) = 2 < 100 =" (100,100).
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