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Let there be four contestants in a game similar to "The Price Is Right". They simultaneously write down bids for an object they don't know, the bids can range from 1 to 1000 USD. The object's value will be 75% of the average of the four bids. The bid nearest to the value will win, and it does not matter whether the bid is too high (unlike in The Price Is Right). Is there an optimal play?

I think there is none, because if there was, then all four contestants would bid the same amount, giving them each a chance of 25% to win. However, for every single contestant it would be advantageous to make a different bid, thus this cannot be a Nash equilibrium.

Is that correct? Or is there a different way to think about this problem?

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    $\begingroup$ Is there not a Nash Equilibrium at bids of one dollar? $\endgroup$ – JP McCarthy Mar 3 '15 at 12:07
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    $\begingroup$ If they all bid the same, it is not advantageous for any ONE person to change their bid because that person will never be closest to the average. $\endgroup$ – Michael Burr Mar 3 '15 at 12:22
  • $\begingroup$ @MichaelBurr but they have to get closest to 75% of the average... therefore this is not a Nash Equilibrium. E.g. if they all bid €100 then if one contestant drops to €99 then they win as they are closest to €75. $\endgroup$ – JP McCarthy Mar 3 '15 at 14:18
  • $\begingroup$ You're right. That makes sense. $\endgroup$ – Michael Burr Mar 3 '15 at 14:19
  • $\begingroup$ So is all four contestants bidding 1 USD the only Nash equilibrium? $\endgroup$ – user220658 Mar 3 '15 at 14:43
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This game is actually pretty common and well-studied by game theorists and behavioral economists. Usually it is called The Beauty Contest, though you'll hear a variety of names, (for instance Guess 2/3 of the average).

If the goal is to guess a fraction $p$ of the average where $p \in (0,1)$, as we have here, then, as several have commented, the unique Nash equilibrium is (generally -- there are exceptions in some specifications of the game where there is rounding and such) for everybody to bid the minimum bid. This is sometimes called a "race to the bottom."

This game is often used in explaining models of Level-k reasoning (sometimes called Cognitive Hierarchy Theory). It's interesting for behavioral economists because how people play depends on their beliefs regarding the sophistication of other players. So PhD economics students, for instances, tend to pick far lower guesses (including the min. guess) than a group of players in which each player is less convinced of the sophistication of others. There are also papers that look at what happens when players repeatedly play this game. As you might imagine, the average guess goes down and down, eventually reaching the minimum guess. Some model this as a learning process.

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    $\begingroup$ I played the 2/3 of the average game with a class of ordinary level maths students. A 'bid' of 18 won which was closest to two thirds of the average which ended up at 18.92. $\endgroup$ – JP McCarthy Mar 3 '15 at 16:17
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    $\begingroup$ @JpMcCarthy Yes - I think an economics class I did it in came to an average of about that also. A common misinterpretation of the game is that people will say that rational players will all choose zero. This is of course incorrect. A rational player who believes others are irrational, or even a rational player who believes that others believe that he is irrational, or any iteration of this logic, may well want to bid above the minimum. Only if all players are rational and there is common knowledge that all are rational is it necessarily optimal to bid the minimum. $\endgroup$ – Shane Mar 3 '15 at 16:50

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