Game theory: competitive auction for the money in a chest

[Master1022]

Homework Statement

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

Relevant Equations

Game theory

Hi,

I am back yet again with another problem I was reading the following question and attempting it. It was an interview problem, so it isn't technically homework, but I don't know where else to post it. I think there are elements of game theory involved, but I have no academic background in the area, so would appreciate any pointers if you think it would be useful.

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

Attempt:
For (a), the question says no more about what our aim is. If I assume that the aim is to maximize our profit and not ever make less than or equal to our opponent, then I would just choose 51 each time. That way, the only way they can get a payout is if they choose $\leq 49$.

However, if the goal was just to maximize my own profit, then I suppose I could just choose $50$ each time so as to create a favorable situation for both players such that we can each maximize our outcomes.

Does this seem reasonable for part (a)?

I am still thinking about part (b)

Thanks in advance.

 

[.Scott]

There are different assumptions that can be made about your opponent and about your objective.

First, the other player isn't referred to as an "opponent" until part b. So I will call them a "game mate".

Second: The rules do not say that you are playing this multiple times. On the face of it, you and your game mate get only one shot at this. If you do get multiple plays, then certainly part of your objective would be to get in as many profitable plays as possible.

You described a competitive situation.

If your objective is to never loose to your game mate, then pick 50 - you may win or tie, but you will never loose.

If your objective is to win as often as possible compared to your game mate, and you don't care about the money, then pick 51 as you suggested.

But if all you want is to win something, then pick 1
- If your game mate is random, you will ave a 99% chance of winning.
- If your game mate is not random, then your chance of wining is unknown. They may decide to pick 100 every time. In any case, chance of winning is depends on your opponents inclinations.

If you want to maximize you expected winning:
- If your game mate is random, and you pick "n", then you will win n a portion (101-n)/101 of the time. So you want to maximize n(101-n)/101, so n=50.5 - you get the same average expected result ($25.25) with either 50 or 51. So it's your choice based on what you wish for you game mate. 50 would allow them to win as often as you.
- If your opponent is not random, I would pick 50 and (if I could), tell him I was picking 50.

Part B
Part b is interesting because although your game mate is described as an "opponent", a mechanism is provided for cooperation.
I would respond by telling them I was going to play 50 - and then I would play that.

 

[haruspex]

Suppose you choose a value X with pdf p(X). Your opposite number, being equally smart, uses the same distribution.
Your outcome is $\int\int p(x)p(y)x.dxdy$ over the region $x>0, y>0, x+y<100$.
Looks like an exercise in advanced calculus of variations.