### 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:- You make a bid.
- A price is randomly generated.
- 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).

- 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 , where i is the number generated minus 1.
- (100-X)% of the time, you make X.
- Your expected reward is one hundredth of .
- 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:- You make a bid.
- A price is randomly generated.
- 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).

- 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 .
- (100-X)% of the time, you make X.
- Your expected reward is one hundredth of .
- 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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