Game Theory Musing

[barking_iguana]

Suppose you and an opponent each have an equal and large amount of resources. You will each secretly allocate those resources to three contests. For each contest, whoever allocates more resources wins. If each of you allocate the same amount to a contest, a fair coin is flipped to determine that contest. Your object is to win at least two of the three contests. Your opponent is a geek who will sacrifice any chance of guessing your psychology so that it is impossible for you to guess theirs.

How should you allocate your resources?

Comments

[bart-calendar.livejournal.com]

34, 34, 32.

[barking-iguana.livejournal.com]

You have to use some random function to determine the allocations, as well as assigning which allocation goes to which contest. If I knew you were following what you said, I could allocate 50%-50%-0% and beat you every time. But if you knew I was doing that, you could allocate 98%-1%-1% and beat me two times out of three.

[avram]

Does all of the allocation get done in advance? In other words, are my resources for the 2nd and 3rd contests already set before I know who won the 1st?

[barking-iguana.livejournal.com]

Yes. Though "no" would also be interesting, it's not what I meant.

[markgritter.livejournal.com]

Seems like a hard problem to solve without some insight. I don't know how to draw the appropriate multidimensional reaction curve. :)

I tried brute-forcing a solution using fictitious play, and I can get convergence to a pretty good solution when restricting the choices to multiples of 10 or 20. But I can't understand the resulting mixed strategy, for example, with choices limited to 0%, 10%, ..., 100% I get:

(0, 50, 50) 5.08%
(10, 20, 70) 4.38%
(50, 0, 50) 4.24%
(60, 10, 30) 4.19%
(10, 60, 30) 3.95%
(0, 30, 70) 3.88%
(40, 50, 10) 3.82%
(50, 40, 10) 3.73%
(40, 0, 60) 3.72%
[snip]

though some choices such as (80, 10 10) should not be played at all.