The weak bidder has a valuation distributed uniformly on [0, 1], while the strong bidder has a valuation that is independently distributed uniformly on [0, v ¯] for some v ¯ > 1. If v ¯ is very large, then the strong bidder is much stronger than the weak bidder. Let sw(v) denote the equilibrium strategies for the weak bidder, let Pw(v) denote the probability that the weak bidder expects to win in equilibrium, and let Tw(v) denote the expected payment for a weak bidder (i.e. Pw(v) · sw(v)).

1. (a) Let ss(v) denote the strong bidder’s equilibrium strategy. Is ss(v) greater or less than sw(v)?
2. (b) If v ¯ increases, will Pw(v) increase or decrease for 0
3. (c) If v ¯ increases, will sw(v) increase or decrease for v > 0?
4. (d) If v ¯ increases, will Tw(1) increase or decrease? [Hint: Pw(1) = 1 regardless of v ¯.]
5. (e) If v ¯ increases, will Tw(v) increase or decrease for very small values of v? [Hint: sw(v) is already very close to v when v is very small. Since sw(v)

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