Game theory/probability puzzle

Question

Part 1:

You play a game where there is a box with 100$ and there are two players, each of you should write a number 0-100 on paper, then you show your numbers, if the sum is higher than 100 then each of you get 0 dollars, else you get what you wrote. What is your strategy?

Part 2:

You play the same game but your opponent told you that he is putting 80 (he might change his mind) - what is your plan?

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For part 1, I think the Nash equilibria are (my number, opponent's number) = (x, 100 - x) for x between 0 and 100? Is this right, and if so, how to choose between them to pick your move? I was thinking about saying 50 as my number for symmetry/aesthetic reasons but would be interested in how one is supposed to tackle this problem. I drew out a reward matrix and tried iterated removal of weakly dominated strategies to get (50, 50) as the only cell remaining - would be grateful for any thoughts on the validity of this.

For part 2, I thought, if you're allowed to say something to the opponent, you could say that you were going to put 50, to try to force your opponent down to 50? Again I have no idea how to approach this. Any help would be much appreciated.

Answer 1

For the second part, I think I'd say to her "I'll choose the same number as you do". This hints to the other player that she should choose $50$, because both larger and smaller numbers result in lower payments ($0$ if higher, $x<50$ if lower). Of course, this depends on the credibility of your "threat".

Answer 2

The question of choosing a NE among several options is an open question - there is no general mechanism to choose among them (think about Battle of the Sexes, for a very simple example). In many papers, the "punch line" is adding some restriction or addition to the model that ruins other equilibria or somehow helps players choose one. Here, the only symmetric equilibrium is indeed $(50,50)$, but all the $(x,100-x)$ equilibria are also good.

The same goes for the second part. Without ability to credibly commit to $80$, his $80$ is more or less worthless.