Mechanism design requires a solution concept to predict the strategies the agents will select in various circumstances. Knowing these strategies will, in turn, ensure that the properties of a particular mechanism can be predicted. Ostensibly, a mechanism may implementscf(θ)under a wide variety of solution concepts, of which we only provide a few of the most important ones here (see [MasColell et al., 1995] for a more in-depth study). As stated in chapter 1, it is up to the designer to select the appropriate solution concept which is achieved by setting S and
g(s). These design parameters, along with the design environment, will lead to different kinds of solutions arising. For analysis, we can partition games into cooperative and non-cooperative games.
In this thesis we focus on competitive game theory purely because it has been the more re- searched field in terms of mechanism design and is more applicable to the situations which we wish to study (see Chapter 1). We present the three most important solution concepts in com- petitive game theory below (see [Osborne and Rubinstein, 1994] for a fuller treatment). Each of the solution concepts presented require stronger assumptions about agents and are, therefore, a weaker implementation concept (i.e. the confidence with which the equilibrium can be pre- dicted is weaker or the environment in which the implementation is carried out is more restric- tive). Nevertheless, all these solution concepts are based around the notion of a best-response strategy, which is the best strategy to play given the (expected) actions of other agents. These solution concepts relate to strategic games (a.k.a normal form games).
Definition 2.4. Strategic Game. A strategic game is one where each agenti∈ I chooses its strategysi ∈Sibased on its preferences or typeθi ∈Θiwhich then leads to an outcomeo∈ O
determined by the outcome functiong(.). Thus a strategic game is completely defined by the tupleΓ = (I,Θ, S, g(.))
Thus a strategic game is a one-shot game. The agents choose their actions and the outcome function determines the outcome. To this end, within stategic games we define the following three equilibrium solutions:
Definition 2.5. Dominant Strategy. In a dominant strategy equilibrium each agent has a best-
response strategy no matter what strategy is selected by the other agents. Formally, we have:
s∗i(θi) = arg max si
ui(θi, g(si(θi), s−i(θ−i))), ∀s−i,∀θ−i (2.3)
for allθi ∈Θi.
Definition 2.6. ex post Nash. Each agent’s strategy is a best-response to the strategy of other
agents, no matter what their types, as long as they also play an equilibrium strategy:
s∗i(θi) = arg max si
ui(θi, g(si(θi), s∗−i(θ−i))), ∀θ−i (2.4)
for allθi ∈Θi, wheres∗−i(θ−i)denotes the strategies selected by other agents.
Definition 2.7. Bayesian-Nash. Each agent selects a best-response strategy to maximise its
expected utility given its beliefs about the distribution over types:
s∗i(θi) = arg max si Eθ−i[ui(θi, g(si(θi), s ∗ −i(θ−i)))] (2.5) for allθi ∈Θi.
A dominant strategy equilibrium is a very robust solution concept because an agent does not need to form beliefs either about the rationality of other agents or about the distribution over the types of other agents. An example of a dominant strategy implementation is the Vickrey auction. In this auction, the best strategy for an agent is to bid truthfully. This is irrespective of what the other agents bid.
An ex post Nash equilibrium requires common knowledge about the rationality of agents, but does not require any knowledge about type distributions. In this sense, ex post Nash has a no- regret property such that an agent does not want to deviate from its strategy even once it knows the strategies and types of other agents. As a simple example, a straightforward bidding strategy in which an agent bids in each round of an ascending-price Vickrey auction2 to maximise its
utility is an ex post Nash equilibrium [Gul and Stacchetti, 2000; Parkes, 2001].3
2This is a modified Vickrey auction where now the auction is conducted over multiple rounds. At each round, the
results of the previous round are known and the auction ends when there is no change in results over two rounds.
3
This is not a dominant strategy in this relaxed auction because another agent might condition a “crazy strategy” such as “I will bid to $1 million” on the price hitting a particular target value. In this case an agent that would otherwise win should submit a jump bid past this target value.
The weakest solution concept adopted in mechanism design is the Bayesian-Nash equilibrium (BNE). In a BNE an agent must both hold beliefs about the rationality of other agents, and also correct beliefs about the distribution on types of other agents. The first-price sealed bid auction is a classic example with a simple Bayesian-Nash equilibrium. For example, given a symmetric distribution of agent types with values that are identically and independently distributed vi ∼ U(0,1)the symmetric BNE is for agents to plays∗i(vi) = (|I| −1)vi/|I|.
Given these solution concepts, we now focus on what properties we want to emerge from the mechanism when a solution has been reached.