# 11.6 Mechanism Design

The earlier discussion on agents choosing their actions assumed that each agent gets to play in a predefined game. The problem of mechanism design is to design a game with desirable properties for various agents to play.

A mechanism specifies the actions available to each agent and the outcomes of each action profile. We assume that agents have utilities over outcomes.

There are two common properties that are desirable for a mechanism:

• A mechanism should be easy for agents to use. Given an agent’s utility, it should be easy for the agent to determine what to do. A dominant strategy is a strategy for an agent that is best for the agent, no matter what the other agents do. If an agent has a dominant strategy, it can do its best action without the complicated strategic reasoning described in the previous sections. A mechanism is dominant-strategy truthful if it has a dominant strategy for each agent and, in the dominant strategy, an agent’s best strategy is to declare its true preferences. In a mechanism that is dominant-strategy truthful, an agent simply declares its true preferences; the agent cannot do better by trying to manipulate the mechanism for its own gain.

• A mechanism should give the best outcome aggregated over all of the agents. For example, a mechanism is economically efficient if the outcome chosen is one that maximizes the sum of the utilities of the agents.

###### Example 11.18.

Suppose you want to design a meeting scheduler, where users input the times they are available and the scheduler chooses a time for the meeting. One mechanism is for the users to specify when they are available or not, and for the scheduler to select the time that has the most people available. A second mechanism is for the users to specify their utility for the various times, and the scheduler chooses the time that maximizes the sum of the utilities. Neither of these mechanisms is dominant-strategy truthful.

For the first mechanism, users may declare that they are unavailable at some time to force a time they prefer. It is not clear that being available at a certain time is well defined; at some stage, users must decide whether it is easier to reschedule what they would have otherwise done at some particular time, rather than say they are unavailable at this time. Different people may have different thresholds as to what other activities can be moved.

For the second mechanism, suppose there are three people, Alice, Bob, and Cory, and they have to decide whether to meet on Monday, Tuesday, or Wednesday. Suppose they have the following utilities for the meeting days:

Monday Tuesday Wednesday
Alice 0 8 10
Bob 3 4 0
Cory 11 7 6

The economically efficient outcome is to meet on Tuesday. However, if Alice were to change her evaluation of Tuesday to be 2, the mechanism would choose Wednesday. Thus, Alice has an incentive to misrepresent her values. It is not in Alice’s interest to be honest.

Note that, if there is a mechanism that has dominant strategies, there is a mechanism that is dominant-strategy truthful. This is known as the revelation principle. To implement a dominant-strategy truthful mechanism, we could, in principle, write a program that accepts from an agent its actual preferences and provides to the original mechanism the optimal input for that agent. This program would optimally lie for the agent.

It turns out that it is essentially impossible to design a reasonable mechanism that is dominant-strategy truthful. Gibbard [1973] and Satterthwaite [1975] proved that, as long as there are three or more outcomes that are possible to be chosen, the only mechanisms with dominant strategies have a dictator: there is one agent whose preferences determine the outcome. This is known as the Gibbard–Satterthwaite theorem.

One way to obtain dominant-strategy truthful mechanisms is to introduce money. Assume that money can be added to utility so that, for any two outcomes $o_{1}$ and $o_{2}$, for each agent there is some (possibly negative) amount $d$ such that the agent is indifferent between the outcomes $o_{1}$ and $o_{2}+d$. By allowing agents to be paid to accept an outcome they would not otherwise prefer, or to pay for an outcome they want, we can ensure an agent does not gain by lying.

In a VCG mechanism, or a Vickrey–Clarke–Groves mechanism, the agents declare their values for each of the outcomes. The outcome that maximizes the sum of the declared values is chosen. Agents pay according to how much their participation affects the outcome. Agent $i$ pays the sum of the value for the other agents for the chosen outcome minus the sum of the values for the other agents if $i$ had not participated. The VCG mechanism is both economically efficient and dominant-strategy truthful, assuming that agents only care about their utility and not about other agents’ utilities or other agents’ payments.

###### Example 11.19.

Consider the values of Example 11.18. Suppose the values given can be interpreted as equivalent to dollars; for example, Alice is indifferent between meeting on Monday or meeting on Tuesday and paying $8.00 (she is prepared to pay$7.99 to move the meeting from Monday to Tuesday, but not $8.01). Given these declared values, Tuesday is chosen as the meeting day. If Alice had not participated, Monday would have been chosen, and so the other agents have a net loss of 3, so Alice has to pay$3.00. The net value to her is then 5; the utility of 8 for the Tuesday minus the payment of 3. The declarations, payments, and net values are given in the following table:

Monday Tuesday Wednesday Payment Net Value
Alice 0 8 10 3 5
Bob 3 4 0 1 3
Cory 11 7 6 0 7
Total 14 19 16

Consider what would happen if Alice had changed her evaluation of Tuesday to 2. In this case, Wednesday would be the chosen day, but Alice would have had to pay \$8.00, with a net value of 2, and so would be worse off. Alice cannot gain an advantage by lying to the mechanism. One way to think about the payment is that Alice needs to bribe the mechanism to go along with her favourite choice.

One common mechanism for selling an item, or a set of items, is an auction. A common auction type for selling a single item is an ascending auction, where there is a current offering price for the item that increases by a predetermined increment when the previous offering price has been met. Offering to buy the item at the current price is called a bid. Only one person may put in a bid for a particular price. The item goes to the person who put in the highest bid, and the person pays the amount of that bid.

Consider a VCG mechanism for selling a single item. Suppose there are a number of people who each put in a bid for how much they value an item. The outcome that maximizes the payoffs is to give the item to the person who had the highest bid. If they had not participated, the item would have gone to the second-highest bidder. Therefore, according to the VCG mechanism, the top bidder should get the item and pay the value of the second-highest bid. This is known as a second-price auction. The second price auction is equivalent (up to bidding increments) to having an ascending auction, where people specify how much they want to pay as a proxy bid, and there is an agent to convert the proxy bids into real bids. Bidding in a second-price auction is straightforward because the agents do not have to do complex strategic reasoning. It is also easy to determine a winner and the appropriate payment.