Let's break down Problem 1.4 step by step: --- ## Problem 1.4 Overview - **10 players** (risk-neutral), each with a private value \( v_i \sim \text{Uniform}[,1] \). - **Auction Rule:** Highest bidder wins, but pays the **average of the 2nd and 3rd highest bids** (unless otherwise specified in later parts). --- ## (a) **Bidding \( v_i \) is a Weakly Dominant Strategy? Is the Auction Efficient?** ### **Weak Dominance** - **Definition:** A strategy is weakly dominant if, for any strategies of other players, it never does worse and sometimes does better than any other strategy. - **Direct Revelation (Truthful Bidding):** In this auction, your bid only affects whether you win. If you win, your price **does not depend on your own bid** (but on the 2nd and 3rd highest bids). - **Reasoning:** - If you bid above your value, you risk winning and paying more than your value. - If you bid below your value, you might lose when you would have gotten positive utility. - **Conclusion:** **Bidding \( v_i \) is a weakly dominant strategy**, as in a Vickrey (second-price) auction. ### **Efficiency** - **Efficient Allocation:** The good should go to the player with the highest value. - Since each player's optimal strategy is to bid their true value, the highest value always wins. - **Conclusion:** **Auction is efficient**. --- ## (b) **Revenue Equivalence Theorem (RET)?** - **RET:** If bidders are risk-neutral, have independent private values, and the allocation rule is the same, then all standard auction formats yield the same expected revenue. - **Does RET Apply Here?** - This is a special auction: winner pays the average of 2nd and 3rd highest bids (not a standard auction). - In the standard 1st-price auction, winner pays the highest bid among the others; in 2nd-price, pays the 2nd highest. - **The allocation rule is the same** (highest value wins), but **payment rule is different**. - **RET applies** as long as payment depends only on the ranking and not on the actual bid. - **Conclusion:** **RET will apply**; expected revenue is the same as 2nd-price (Vickrey) auction. --- ## (c) **Seller’s Equilibrium Expected Revenue** - **Truthful Bidding:** Each \( v_i \sim \text{Uniform}[,1] \). - Winner pays the **average of the 2nd and 3rd highest values**. ### **Expected Value of Order Statistics** Let \( X_{(k)} \) be the \( k \)-th highest among \( n \) i.i.d. uniform \([,1]\) random variables. - \( E[X_{(k)}] = \frac{k}{n+1} \) So for \( n=10 \): - 2nd highest: \( E[X_{(9)}] = \frac{9}{11} \) - 3rd highest: \( E[X_{(8)}] = \frac{8}{11} \) - **Expected payment:** \( \frac{1}{2}(E[X_{(9)}] + E[X_{(8)}]) = \frac{1}{2}\left(\frac{9}{11} + \frac{8}{11}\right) = \frac{17}{22} \) **Conclusion:** **Seller's expected revenue = \( \frac{17}{22} \approx .773 \) units.** --- ## (d) **What if Utility is \( u_i(z) = z^r \), \( < r < 1 \)?** - **Non-linear utility:** Players are **risk-averse**. - In a **2nd-price auction**, truth-telling is still a weakly dominant strategy for risk-neutral players. But with risk aversion, players may shade bids. - **Revenue Equivalence Theorem no longer holds**: With non-linear utility, the payment rule influences bidding behavior and thus expected revenue. - **Conclusion:** **Seller's expected revenue can be higher in 2nd-price auctions with risk-averse bidders, since they tend to bid higher.** --- ## (e) **What if Winner Pays Average of Top Three Bids?** - **Now, winner pays \((\text{Bid}_1 + \text{Bid}_2 + \text{Bid}_3)/3\)** - **Strategic Implications:** - Bidding your true value is **no longer weakly dominant**. - Your own bid now affects the price you pay if you win, so you may shade your bid below your value. - **Inefficiency** can arise: highest-value player might not win, especially if strategic bid shading is significant. - **Seller's Expected Revenue:** - Since players shade their bids, expected revenue is **lower** than in (c). --- ## **Summary Table** | Auction Format | Efficient? | Truthful Bidding? | RET Holds? | Expected Revenue | |------------------------------------------|------------|-------------------|------------|------------------------| | Avg(2nd, 3rd) price (risk neutral) | Yes | Yes | Yes | \( \frac{17}{22} \) | | 2nd price (risk-averse) | Yes | No | No | Higher than risk-neutral| | Avg(Top 3) price (risk neutral) | No | No | No | Lower than above | --- If you need detailed derivations for order statistics or strategic analysis for the average of top three case, let me know!

To find the geometric mean rate of increase of the cost of a 30-second commercial for the Super Bowl over the years, we can use the following formula: ### Geometric Mean Formula The geometric mean \( r \) over \( n \) periods is given by: \[ r = \left( \frac{V_f}{V_i} \right)^{\frac{1}{n}} - 1 \] Where: - \( V_f \) = final value (cost in 2021) - \( V_i \) = initial value (cost in 1967) - \( n \) = number of years between the two values ### Given Values - \( V_i = 45500 \) (cost in 1967) - \( V_f = 5600000 \) (cost in 2021) - \( n = 2021 - 1967 = 54 \) ### Calculation Steps 1. **Calculate the ratio of final to initial value:** \[ \frac{V_f}{V_i} = \frac{5600000}{45500} \approx 123.076 \] 2. **Apply the geometric mean formula:** \[ r = \left( 123.076 \right)^{\frac{1}{54}} - 1 \] 3. **Calculate the \( 54^{th} \) root:** Using a calculator or software, \[ \left( 123.076 \right)^{\frac{1}{54}} \approx 1.06357 \] 4. **Subtract 1:** \[ r \approx 1.06357 - 1 \approx 0.06357 \] 5. **Convert to percentage:** \[ r \approx 0.06357 \times 100 \approx 6.36\% \] ### Conclusion The geometric mean rate of increase for the cost of a 30-second commercial between 1967 and 2021 is approximately **6.36%** per year.