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I am working on the exercise $4.4$ of Game Theory: an Introduction by Steven Tadelis.

The Chapter $4$ is about Beliefs, Best-Response Correspondences and Rationalizability in Game Theory.

I am unable to understand how to approach the exercise:

eBay’s Recommendation: It is hard to imagine that anyone is not familiar with eBay, the most popular auction web site by far. In a typical eBay auction a good is placed for sale, and each bidder places a “proxy bid,” which eBay keeps in memory. If you enter a proxy bid that is lower than the current highest bid, then your bid is ignored. If, however, it is higher, then the current bid increases up to one increment (say, $0.01\$$) above the second highest proxy bid. For example, imagine that three people have placed bids on a used laptop of $55\$$, $98\$$, and $112\$$. The current price will be at $98.01\$$, and if the auction ended the player who bid $112\$$ would win at a price of $98.01\$$. If you were to place a bid of $103.45\$$ then the player who bid \$112 would still win, but at a price of $103.46\$$, while if your bid was $123.12\$$ then you would win at a price of $112.01\$$.

Now consider eBay’s historical recommendation that you think hard about the value you impute to the good and that you enter your true value as your bid—no more, no less. Assume that the value of the good for each potential bidder is independent of how much other bidders value it.

a. Argue that bidding more than your valuation is weakly dominated by actually bidding your valuation.

b. Argue that bidding less than your valuation is weakly dominated by actually bidding your valuation.

c. Use your analysis to make sense of eBay’s recommendation. Would you follow it?

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a) What happens if you bid more than your valuation and win? So if you have something at 98 and bid 98.01, you have utility -0.01 < 0 if you win. So how can you improve your outcome here?

b) Suppose you have valuation $v$ and bid $b < v$. What happens if someone else bids $b^{\prime} \in (b, v)$? How could you improve your outcome?

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