|Sep23-12, 01:52 PM||#1|
A "game" involving limits and the epsilon-delta concept
I have a question about a "game" I've thought up and I want to know if my logic is right, because if I'm wrong I fear it means i don't quite grasp the epsilon-delta concept that is used to examine limits, derivatives, continuity, etc., so much more.
So say we have a player P, and it's opponent, O. Now, P is given two options:
1) Take a lump sum of money, $K, and walk away. (O will also get $K is this option is chosen)
2) Play a game with O and take the winnings, $WP from that game. (O will also get some winnings, $WO)
This is how the game works. P picks a number, ε>0, which can be arbitrarily small. In response, O is allowed to pick a number, δ>0, which can be arbitrarily small. Once the selections are made, the following winnings WP and WO are given to the players:
WP = ε(x-L) + δ(L-y) + K (payoff given to P)
WO = ε(L-x) + δ(y-L) + K (payoff given to O)
But say that: L>x, L>y
So if we think about this, we can see that it's in P's best interest to pick the smallest ε possible, since the amount (x-L) is negative (because L>x) and it wants to minimize this amount. Similarly, O wants to pick the smallest δ possible, since the amount (y-L) is negative and it wants to maximize it's profit as well. Also I notice that unfortunately these selections have the opposite effect on the other player, the smaller value of epsilon that P selecy ts, this makes O's winnings smaller. Similarly, the smaller delta that O picks, this makes P's winnings smaller.
Now the question is, is it ever rational for P to play the game, or should it just take the lump sum of K? To make this decision I feel we can look at the winnings, WP, that P will get if it decides to play, and compare it to the lump sum, K it can chose to take. If WP > K, then P should play:
WP = ε(x-L) + δ(L-y) + K
WP = lim(ε-->0)+[ε(x-L)] + lim(δ-->0)+[δ(L-y)] + K
WP = 0 + 0 + L
WP = K
As we take these limits arbitrarily close to 0, clearly the value of those limits is 0. So it seems like in either case, P will make the same amount.
But here's the part I'm confused about. P and O both still have to chose some value, epsilon and delta, right? So even though they want to minimize their epsilon and delta, you'll still have these tiny, infinitesimal, negligible amounts and the actual winning will really be:
WP = c1 + c2 + K
where c1 is the tiny negative amount that P is trying to minimize and c2 is a tiny positive amount that comes as a result of O's selection of δ. Now, let's assume O has no vendetta against P, so it has no reason to try and hurt it. So it's not going to go out of it's way to pick a δ smaller than P's ε. But still, both are trying to make these values as small as possible. So it seems to me like P would only want to play this game if it could gaurantee that |c1| < c2, but since it doesn't get to chose ε in response to O's δ, there's no way to ensure this. So P should never play the game because it risks a situation where O selects a δ < ε which would mean it's winnings, WP < K (and if it choses to just take the lump sum of K it would fair better in that case).
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