Impressions on Nagel (1995)

Title: Unraveling in Guessing Games: An Experimental Study Link to the paper.

Authors: Nagel

Year published: 1995

Journal: American Economic Review

This paper shows that the beauty contest game is very well-suited for investigating \(k\)-level reasoning of players

Today's snarky review is not really going to be snarky nor will it follow the same format of the usual summaries. I first encountered this game in 2006, through one of our professor's demonstrations in a Young Economists' Convention. I knew the answer, having heard about it from someone else before, though I didn't know why the answer was so. It is worth noting that we didn't have proper game theory in our university back then - and for many years after that, I had trouble with the notion of "strategic thinking". It was very hard to accept that agents regularly attempted to outsmart each other until there is no more outsmarting to be possibly done. This is probably how they should explain "equilibrium strategy" to first-timers.

Anyway, fast forward to many years later and I ended up doing work on theoretical beauty contests myself. There is quite a huge gap in the techniques used to find equilibrium for theoretical beauty contests. The reason is that the typical solution method was to "assume some form, and verify that your assumption is correct". This was also very hard to process - yet many fields in microeconomic theory use this method. It is nice to see the very first paper that shows how the train of thought of "rational players" should go.

The game (verbatim)

A large number of players have to state simultaneously a number in the closed interval \([0,100]\). The winner is the person whose chosen number is closest to the mean of all chosen numbers multiplied by a parameter \(p\), where \(p\) is a predetermined positive parameter of the game; \(p\) is common knowledge.

The payoff to the winner is a fixed amount independent of the stated number and \(p\). If there is a tie, the prize is divided equally among the winners. The other playes whose chosen numbers are further away receive nothing.

How to win this game

• In principle it is enough to guess the average and apply \(p\) to it
• But this is where things get tricky. The average excludes one from choosing the high numbers, because then it would be too far from the average
• And the average would depend on what the opponents were thinking
• Suppose everyone else chose randomly within \([0,100]\). Then, the average of their answers would be around 50. To win in this case, you should pick \(50\cdot p\). This is what is called level-1 thinking.
• Suppose everyone else was like you. Then they would each pick \(50\cdot p\). Then the average would be \(50\cdot p\). Then you should pick a number that is \(p\) times the average - namely \(50\cdot p^2\). This is called level-2 thinking.
• Suppose everyone else thought up to this point. Then they would each pick \(50\cdot p^2\). Then the average would be \(50\cdot p^2\). Then you should pick a number that is \(p\) times the average - namely \(50\cdot p^3\). This is called level-3 thinking.
• By this point it becomes obvious that you and your opponents could do this as many times as possible. This is generally called level \(k\) thinking.
• But again you just need to be one step ahead of what the opponents are thinking. So if you think your opponents are unsophisticated, it is enough to choose \(50\cdot p\).

The equilibrium in theory

For \(p\in[0,1)\), there exists only one Nash equilibrium, at which all players choose 0. Why?

• Let's take for a moment how the average is calculated. For \(n\) items \(x_1,\cdots, x_n\), the average is \(m=\sum_{i=1}^n \frac{1}{n} x_i\)
• Among the \(x_i\), there is one highest number. Let us call this number \(k\). But the player who chooses \(k\) will never win because \(k>m>p\cdot m\).
• So if this player is rational, he will never choose \(k\). He will choose something lower than \(k\).
• Now assume all the players are rational. Then they will realize that they can keep choosing lower numbers until there is only the lowest possible number is zero.
• It might be worth noting that one can only deviate downwards not upwards.

What level did people actually get to in Nagel's experiments?

• People seemed to be influenced by the \(p\) that was set by the experimenter.
• Each successive round, the behavior moves in a consistent behavior towards equilibrium
• Some sessions converge faster to zero than others.
• Bounded rationality (how actual people make decisions) seems to be quite different from game theoretic reasoning.