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?
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.