# Demand-Response Control in Smart Grids
## 6 Theoretical and Experimental Comparison between Vickrey and Dutch Auctions
### Comparison between Vickrey and Dutch Auctions Based on a Case Study  
We aim to compare the outcomes of both Vickrey and Dutch auctions. To do that, we take a case study of a provider selling electricity using an auction approach. The provider is looking to sell a total of 600 Kw.  
Let us suppose five consumers (A, B, C, D, and E) are bidding on electricity.  
* Consumer A bids 300 Kw at $400
* Consumer B bids 100 Kw at $500
* Consumer C bids 400 Kw at $100
* Consumer D bids 150 Kw at $300
* Consumer E bids 200 Kw at $200
* In a Dutch Auction Sorted list of consumers established with the highest price at the top: \(\bigcirc\) Consumer B: 100 Kw at $500 (600 \(-\) 100 = 500 Kw remaining) \(\bigcirc\) Consumer A: 300 Kw at $400 (500 \(-\) 300 = 200 Kw remaining) \(\bigcirc\) Consumer D: 150 Kw at $300 (200 \(-\) 200 = 0 Kw remaining) \(\bigcirc\) Consumer E: 200 Kw at $200
* Consumer C: 400 Kw at $100  
In this example, the electricity would be priced at $300 per Kw, because the last bid of 150 Kw at $300 filled out the total number of Kw that the provider is looking to offer. The final price is $300, and the three best bidders (B, A, and D) pay the final price of $300 (100 Kw for B, 300 Kw for A, and 150 Kw for D) instead of their initial bids of $500, $400, and $300, respectively. Consumers E, F, and C would not receive anything because the number of Kw is already filled. The consumers A, B, and D are satisfied with paying $300 per Kw, because their bids were higher. The consumers C and E cannot purchase any electricity because their bid price was too low, and there is no electricity left to purchase.
* In a Vickrey Auction Sorted list of consumers established with the highest price at the top: \(\bigcirc\) Consumer B: 100 Kw at $500 \(\bigcirc\) Consumer A: 300 Kw at $400 \(\bigcirc\) Consumer D: 150 Kw at $300 \(\bigcirc\) Consumer E: 200 Kw at $200 \(\bigcirc\) Consumer C: 400 Kw at $100  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Criterion** & **Vickrey Auction** & **Dutch Auction** \\ \hline Number of messages & N\({}_{\text{Msg}}\) = 2\(\mathcal{\text{C}}\)n + 1 & N\({}_{\text{Msg}}\) = Cn(B + 1) + 1 \\ \hline The price is determined by the agent itself & Agent’s bid determines whether or not he wins the auction, but not the price & Agent’s bid determines the price \\ \hline Vulnerability to & Collusive agreement & The consumers are motivated to break the collusion agreement \\ bidder collusion & for Vickrey & The auctioneer can overstate \\ \hline Vulnerability to a & 2 nd highest bid to the lying auctioneer & The auctioneer cannot lie because bids are public \\ \hline \hline  & & Dutch may not reveal this info as accurately \\ Undesirable private information revelation & cost information revealed because truthful bidding is a dominant strategy in Vickrey & No dominant strategy \\  & Agents’ strategic marginal & \(\bullet\) & Bidding decisions depend on beliefs about others \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison between Vickrey and Dutch auctions.  
It is evident that consumer B has the highest price ($500), winning the auction bid but paying the price of the second-highest bid (A), which is $400.
* Comparison between Dutch and Vickrey Auctions The parameters we need to compare are the final price of the item, the amount of money paid by each bidder, and the amount of electricity each bidder receives in return. Table 6 shows that in a Dutch auction, 80% of consumers (A, B, and D) receive the requested electricity based on their respective bids and pay in return the same amount ($300), whereas in a Vickrey auction, only one bidder (B) receives the electricity needed. In Dutch auctions, the provider is also satisfied, as it sells all the electricity that it provides at a reasonable price; here, the total amount received will be $570,000. Whereas, in the Vickrey auction, only a small part of the provided electricity is sold. This demonstrates that the Dutch auction produces a more equitable outcome than the Vickrey auction for both consumers and provider.