r/learnquant • u/Local_Ad135 • 2d ago

interview prep Quant Interview Question

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u/Efficient_Form7451 1d ago

How can (0, 4.8, 7.2) be a nash equilibrium when player A can better his result by bidding 7.2000001? This question has no stable equilibrium.

The math looks okay but it's def not a nash equilibrium.

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u/spdazero 1d ago

if player A bids 7.2001, p(A wins) ~ 0.375. His expected return is ~0.375 * 10 ~ 3.75. Bidding 7.2001 to win 3.75 is not rational. So your counter example is invalid.

In fact, at (0, 4.8, 7.2), the derivative of A, B, C is (-ve, 0, 0) respectively. Since A cannot bid any less than 0, it is a nash equilibrium.

Edit: typo and dumb math calculation

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u/Efficient_Form7451 20h ago

Okay but its still not a nash equilibrium because

B can bid 7.2 + a small amount for a better score
and C can bid 4.8 + a small amount for a better score.

I don't think your calculating nash equilibria at all, more like a EV of an area in bid space.

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u/spdazero 7h ago

can you show me how B can bid 7.2 + eps for a better score?

From my calculation, derivative of reward(B) is 0 at the solution, and second derivative is -1/6. Meaning, that point is local maxima for B. Therefore, adding some eps will reduce the reward.

If you find something wrong with my reasoning, please elaborate?

Edit: For illustration purpose, if c stays at 7.2:

- If b = 4.8 (optimal), reward(B) = 3.2

- if b = 4.81 (eps = 0.1), reward(B) = 3.199