June 15, 2012

Experiments: Game Theory

Follow-up to To ensure that you are honest… from Midgley, Christopher - Pointless twaddle and meaningless diatribes

Realised I could model the previous two cases of product purchase as games. So let’s do that.

Game 1:
  1. You make a bid.
  2. A price is randomly generated.
  3. If the price is below your bid, you pay the price and get the item; otherwise you pay nothing (and don’t get the item).
Model as a game:
  • Price to play the game is X
  • You consider the product worth Y
  • We assume the RNG creates numbers between 1 and 100 inclusive.
  • X% of the time, you make Y+i, where i is the number generated minus 1.
  • (100-X)% of the time, you make X.
  • Your expected reward is one hundredth of (100-X)X+\sum_{i=0}^{X-1}{Y+i}=-\frac{X^2}{2}+XY+\frac{199X}{2}.
  • For payoff we subtract the bid.

At 0, we make nothing. At 100, we make 49.5+Y-X. Our maximum here occurs at X=Y+1/2 which, as we’re limited to integer solutions, still returns the same result.

Game 2:
  1. You make a bid.
  2. A price is randomly generated.
  3. If the price is below your bid, you pay your bid and get the item; otherwise you pay nothing (and don’t get the item).
Model as a game:
  • Price to play the game is X
  • You consider the product worth Y
  • We assume the RNG creates numbers between 1 and 100 inclusive.
  • X% of the time, you make Y.
  • (100-X)% of the time, you make X.
  • Your expected reward is one hundredth of XY+(100-X)X=100X+XY-X^2.
  • For payoff we subtract the bid.

At 0, we make nothing. At 100, we make Y-X. At Y=X, we make X, which is our best.

So in conclusion we have that both have the same maxima, but the former has a higher payoff, which I suppose you’d expect.


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