implements the choice rule, i.e. that leads to the same outcomes as the ones determined by the choice rule for each preference profile.
When solving an implementation problem, the planner has to take into account how the players will play any possible game. This means that given a preference profile and a strategic game form, the planner wants to predict how the players will behave in the process of playing the game. Here, she has to make some assumptions about which strategies the players adopt given their preferences. This means that the planner assumes that the players are playing according to some solution concept.
Definition 5.4 (Solution Concept). A solution concept for the environment
hN, C,P,Giis a function S:G × P →2A1×A2×...×An
Finally, we can formalize what it means to implement a choice rule.
Definition 5.5. Let hN, C,P,Gi be an environment and let S be a solution concept. The game formG∈ G with outcome functiong is said toS-implement the choice rulef :P →Cif for every preference profile¹∈ P we haveg(S(G,¹
)) =f(¹). Then we say that the choice rulef isS-implementable inhN, C,P,Gi
If f is S-implementable, this means that assuming that all the players play according to S, then the planner can set the rules of the game in such a way that for every preference profile of the players, their play is also in accordance with the choice rule.
5.2
Mechanism Design in the Framework of
Multi-agent Systems with Preferences
Now, the next step is to investigate mechanism design in the framework of multi-agent systems with preferences. Recall that the job of the planner is to design the rules of a game in such a way that for every preference profile under consideration the possible outcomes will always be exactly those that the choice rule assigned to the preference profile. The planner has to pick from a set of game forms the one that assigns outcomes to action profiles in such a way such that if the players play in accordance to a certain solution concept, then no matter what their preferences are, the outcomes of the game will be as specified by the choice rule.
Now, the task is to transfer this to our framework of multi-agent systems. The idea is the following. After specifying her choice rule, the planner has to pick from a set of multi-agent systems that all have the same states but different accessibility relations. She has to do it in such a way that for the system she picks the following holds: Assume that the agents choose their actions according to a certain solution concept. Then for each preference profile under consideration, the agents will always act such that the possible next states the system can move into as a result of the actions of the agents are exactly those selected by the choice rule for the preference profile under consideration.
58 CHAPTER 5. MECHANISM DESIGN
One thing that is important to note at this point is that in the multi-agent systems that we defined, one effect of the modular approach is that the effects of actions are independent of who exactly performs them. The effects of actions are determined in the environment module which is independent of the agents. If some action is performed, it has a certain effect which does not depend on who performed the action.
In the case of strategic game forms, this independence is only given if the outcome function is such that it assigns the same outcome to action profiles if taking the union of the actions results in the same sets, i.e
g(hA1, . . . , Ani) =g(hB1, . . . , Bni) if [ i∈Ag Ai= [ i∈Ag Bi.
Let us now formalize our ideas of mechanism design for multi-agent systems. As we already mentioned, the planner is given a set of multi-agent systems that only differ in their accessibility relation, i.e. they have the same set of states propositional valuations, the same set of agents an agents have the same abilities with respect to which actions they can perform.
Definition 5.6 (Static Multi-agent System). A static multi-agent system is a tuple hS, Ac,Φ0, π, Ag,acti, where S is a set of states, Ac a set of atomic actions, Φ0 a set of propositional variables, π a propositional valuation, Ag a set of agents andacta functionact:Ag→2Ac assigning sets of actions to each
agent.
So, a static multi-agent system is just a multi-agent system without accessibility relation.
We say that a multi-agent systemM is based on a static multi-agent system ¯
M if it extends it by an accessibility relation.
Now, the environment the planner is working in is a set of multi-agent systems based on the same static multi-agent system, i.e. they only differ in their acces- sibility relations. Then the planner’s task is to pick one of them that implements the desired choice rule. A choice rule assigns to each pair of preference profile and state a set of states that the planner wants the system to move into in the next transition.
Definition 5.7 (Choice Rule for Multi-agent Systems). Given a set of prefer- ence profilesP and a multi-agent systemM =hS, Ac,(→)A⊆Ac,Φ0, π, Ag,acti,
a choice rule for multi-agent systems is a functionf :P ×S →2S.
Next, we define a multi-agent analog of solution concepts as we presented them in the previous section. Given a multi-agent system, a preference profile and a state in the system, a solution concept gives us a set of concurrent actions. Definition 5.8 (Solution Concept for Multi-agent Systems). Let M be a set of multi-agent systems based on a static multi-agent system M¯ =
hS, Ac,Φ0, π, Ag,acti and let P be a set of preference profiles of the agents in Ag over the set of states S. Then a solution concept for M is a function
S:P ×M×S→22Ac
.