Archive discussion: truel
Timothy Chow
The file in the archive that I'm talking about is
http://einstein.et.tudelft.nl/~arlet/puzzles/sol.cgi/decision/truel
I should confess that I have not read the entire file closely yet.
However, there are a few points I want to make.
1. With some difficulty, I tracked down the paper that argues that "everyone
passes" is optimal if and only if N = 3.
D. E. Knuth, "The triel: a new solution," Journal of Recreational
Mathematics 6 (1973), 1-7.
However, the argument given there is unsatisfactory. I corresponded
briefly with Knuth about it and he agreed that this paper is seriously
flawed. (Note the spelling "triel.")
2. There is a brief discussion of the truel in one of Martin Shubik's
books on game theory (I think it's _Game_Theory_in_the_Social_Sciences:_
_Concepts_and_Solutions_) but he does not attempt to give a complete
analysis of the problem.
3. Much of the confusion surrounding this problem arises from the implicit
assumption that an "optimal" strategy for each player exists and is
unique. For N > 2 this is not at all obvious and may not even be true.
Any careful discussion must begin with a precise definition of optimality
and proofs of existence/uniqueness of optimal strategies; otherwise this
tacit assumption can slip in in subtle ways to produce apparent paradoxes.
I didn't see anyone take this approach in the archive file when I skimmed
it; did I miss something?
--
Tim Chow tc...@umich.edu
Where a calculator on the ENIAC is equipped with 18,000 vacuum tubes and weighs
30 tons, computers in the future may have only 1,000 vacuum tubes and weigh
only 1 1/2 tons. ---Popular Mechanics, March 1949
Tim Firman
Timothy Chow wrote:
>
> The file in the archive that I'm talking about is
>
> http://einstein.et.tudelft.nl/~arlet/puzzles/sol.cgi/decision/truel
I skimmed the archive as well. It seemed to (de?)generate a lot
of different sub-puzzles using different starting conditions...
I think, given a zero utility for infinite shooting or death and
positive for winning, and otherwise using the original rules, the
outcome would be A misses, B kills C, giving a 30% A win and 70%B win.
You'd have to specify more completely what your objections are to get
more discussion from me... Certainly the archived discussion was not
very rigorous; it's hard to be rigorous when the starting conditions
are ill-defined or changing.
Tim
Timothy Chow
In article <32B5B7...@lombardi.chem.wisc.edu>,
Tim Firman <fir...@lombardi.chem.wisc.edu> wrote:
>
>I skimmed the archive as well. It seemed to (de?)generate a lot
>of different sub-puzzles using different starting conditions...
>
>I think, given a zero utility for infinite shooting or death and
>positive for winning, and otherwise using the original rules, the
>outcome would be A misses, B kills C, giving a 30% A win and 70%B win.
>
>You'd have to specify more completely what your objections are to get
>more discussion from me... Certainly the archived discussion was not
>very rigorous; it's hard to be rigorous when the starting conditions
>are ill-defined or changing.
I'm not sure exactly what you mean by "starting conditions," but let's
say we have probabilities .9, .7, and .5; that the players rotate clockwise
with a randomly determined player starting; that alliances are not allowed;
that payoff is 1 for winning the duel with no other survivors and 0 otherwise;
and that players *do* have memories. Is this well-defined enough?
Now my question is, what does it mean for a strategy to be "optimal"? (For
example, a typical necessary condition for an optimal solution is that it be
a Nash equilibrium.) Given this definition of "optimal," does each player
have an optimal strategy? Is it unique?