Game Theory and Strategy

The Rationality of Irrationality

Kshitiz Joshi

Published in

Intellectually Yours

5 min readDec 31, 2020

Why do we take not-so-rational decisions sometimes?

(Credits- https://www.cs.virginia.edu/~robins/The_Travelers_Dilemma.pdf)

The “Traveler’s Dilemma” was a problem formulated by Kaushik Basu in 1994, in order to challenge the existing concepts in Game Theory. As Basu suggested in the Traveler’s Dilemma, players attempt to maximize their own payoff, without regard for the other. This game is a demonstration that if people think “irrationally” — as opposed to theories in a conventional understanding — they might have a better chance of increasing their pay-offs.

The game presents a scenario in which an airline severely damages identical antiques purchased by two different travelers. The airline is going to compensate for their loss, but the manager is unaware of the actual value of the goods and estimates it to be in between $2 and $100. She asks both of the travelers to write the worth of those damaged antiques (after separating them, of course) from the given range.

The manager puts forth a condition, if both travelers write down the same amount, that amount will be reimbursed to both of them. If both of them write different numbers, the airlines will consider the lower figure as the actual price of the antique. Thus, both the travelers will be compensated with the lower price along with a bonus of $2 for the person who wrote the lower price for honesty, and a deduction of $2 from the other one for dishonesty.

One might think that the traveler’s choice should be $100 as it is the maximum allowed price. Whereas the Nash equilibrium condition for this problem is only $2 (the minimum allowed price). Let’s find out why. Traveler A’s first thought would be to quote $100, and the second traveler will also quote the same implying both of them will get $100. But upon thinking a bit further, Player A might think that Player B would also choose 100$ and thus A would be better-off putting down $99. If B puts $100, A would receive $101 ($99 + $2 bonus). However, B being in the same position would also think of jotting down $99. So A changes their mind again and thinks of putting $98 which pays them 100$ ($98 + $2 bonus) which would be better than $99 — in the scenario when both put $99. These cycles of thought and reasoning continue till both of them decide to quote the same amount $2, which is the lowest permissible price. And thus, ($2, $2) is the Nash Equilibrium of this game.

Payoff Matrix

Image credits-https://iq.opengenus.org/content/images/2018/07/TD-payoff.png

But something unusual happens whenever this game is played. Experimentally, most participants select a value higher than the Nash Equilibrium and closer to $100 (corresponding to the better payoffs solution). Even though we know that Game theory gives us the most standard answers for any real life problem, here the result calculated by game theory is completely opposite to the standard one.

One might attribute these results to experimenting with normal people, but even with famous game theorists the same thing happened! Only 10% of the game theorists chose the Nash Equilibrium, around 20% chose the highest value and the remaining 70 % chose between $80 and $90. (Becker, T., M. Carter, and J. Naeve, 2005, “Experts Playing the Traveler’s Dilemma”, Discussion Paper 252, Institute for Economics, Hohenheim University, available at: https://www.uni-hohenheim.de/RePEc/hoh/papers/252.pdf ). What was the reason behind this deviation?

To find out, this game was also played with some variations. Instead of having $2 as the bonus/penalty, the price was increased to $80, thereby increasing the risk of a greater loss. This experiment showed that people tried to choose the other way round this time, i.e they opted for values closer to the Nash equilibrium. The players ultimate goal here became to just get some positive pay off. So, just to be sure of the fact that they do get something, their intuitions lead to low figures and safe play. Hence the Nash Equilibrium became the ‘rational’ choice in this case.

Thus one can answer our question of “Why this deviation?”. Our players final aim is to get a higher pay off and hence their decisions are affected by the goal they have in mind. In the first case both players wanted to get a higher pay off whereas in the latter they just wanted to not go empty-handed. The dilemma is neither in the choice between the rational choice and the optimal choice nor in between logic and intuition. The dilemma lies between getting the highest possible amount or atleast getting something, or in other words, how much should you focus on what your opponent is getting?

Saying that “sometimes rationality is not so rational” is something which can not be proven by Traveler’s Dilemma as this dilemma is not framed properly to draw such a conclusion. Consider yourself in this scenario and you need the other player to lose. Having read this article, you might choose a figure closer to the Nash Equilibrium and the other other player who has no idea about “Game Theory” might choose a higher number like $100. Now whose loss is it? In fact, instead of rationality, it is logic and strategy that drive us to our final decisions and conclusions. The optimal choice in our initial case was to write a higher number in order to achieve the goal of ‘Higher pay-off’ and hence this was the logic driving the players to play something completely opposite to what the Nash Equilibrium would expect of them.

In the case of Traveler’s Dilemma, the intuition is right and only awaiting validation by a better logic. The paradoxes of rationality that plague game theory as well as the steps to the solution remain codified in the Traveler’s Dilemma.