Adverbial clauses

Mechanism design is a field in economics that designs economic mechanisms, or in-centives, towards desired objectives. It inherits the strategic settings in game theory that assumes players behave rationally and focuses on the class of games with private informa-tion. However, instead of finding the equilibrium solutions under a given game structure as traditional game theory, it starts with the outcome that satisfies desired properties and traces back to the causing mechanisms, which are usually functions of players’ private information. For this reason, mechanism design is sometimes referred to as ’reverse game theory.’ Precisely, in this thesis, the acknowledged preferred outcome is forming behaviorally stable platoons, meaning that no individual vehicles in them can gain greater benefits by manipulation, such as providing incorrect information or changing positions.

The analysis hence discusses the existence and formats of the proper mechanisms that can successfully implement such an outcome.

Mathematically, consider a game defined by a tuple (N, Θ, A, χ, U) where

• N = {1, 2, ..., n} is the finite set of players or agents.

• Θ = Θ1 × · · · ×Θn, where Θi is the space of private information for agent i. Each individual’s private information is also called her type. The agent and mechanism designer normally share a common prior distributionφ on Θ.

• A = A1× · · · × A_n is the set of actions, where Ai = {..., ai, ...} is the set of actions available to agent i ∈ N.

• χ is the space of outcomes. An outcome function χ : A → χ connects each joint strategy a to an outcome in χ.

• Each agent has an ex post utility function Ui : χ ×Θ × Rⁿ→ R, which is given by

Ui(χ, θ, p) = Vi(χ, θ) − ti

where

– V_i is the payoff function of agent i, indicating the valuation she obtains under the outcomeχ and the given joint type θ,

– t= t1× · · · ×tn ∈ Rⁿis the vector of transfers. An transfer function t : A → Rⁿ determines a transfer for each agent given joint action a.

The utility function stated above is defined under the quasilinear environment, since the transfers p enter every player’s utilities quasilinearly. The mechanism is constructed by the tuple< A, χ, t >, where chi and t are selected subject to certain constraints. In mechanism design problems, oftentimes the actions equal to sending messages, which are subject to players’ true types. Mathematically, it can be formulated as ai = a(θi), a : Θi → A_i, ∀i ∈ N.

However, if the action is revealing the type itself, that is to say, Ai = Θi, ∀i ∈ N, then the mechanism is called a direct mechanism.

Mechanism design problems normally can be formulated as optimization problems that maximize certain objectives by determiningχ and t under required constraints. Objec-tive functions that are commonly seen include maximizing social welfare,P

ivi(χ(a(θ)), θ), and maximizing revenue, P

iti(a)). If the attainable social welfare under an outcome is maximized, the mechanism is called economic efficient.

Core constraints include incentive compatibility, individual rationality, and budget balance.

They ensure the outcome under the designed mechanism is an equilibrium solution for all players. Under the Revelation Principle (Myerson,1981), any mechanisms that are incentive compatible is equivalent to the one that each agent truthfully reports her type. In this way, all mechanisms can convert into direct mechanisms.

2.4.1 Incentive Compatibility

Depending on the extent that private information is disclosed, there are three classes of incentive compatibility: interim, ex post and dominant strategy incentive compatibility.

Definition 2.4.1. (IIC) A mechanism (a^∗, χ, t) is said to be interim incentive compatible if the

following conditions are satisfied:

E[Vi(χ(a^∗(θ))), θ) − ti(a^∗(θ))|θi] ≥

E[Vi(χ(ai, a^∗−i(θ⁻i)), θ) − ti(ai, a^∗−i(θ⁻i))|θi], ∀ai ∈ A_i, ∀i ∈ N (2.4.1) Here, the resulting strategy a^∗describes the situation when every agent maximizes her expected conditional utility when she knows her own type. Thus, the outcome,χ(a^∗θ)), is a Bayesian Nash equilibrium. Suppose function ˆχ : Θ → χ satisfies ˆχ(θi)= χ(a^∗(θi)), ∀i ∈ N, it is also said that the mechanism (a^∗, χ, t) implements ˆχ in Bayesian Nash equilibrium.

Here, ˆχ is the direct mechanism that complies with the Revelation Principle. Though widely used in applications of mechanism design problems, the IIC mechanism can be easily sabotaged once there is information leakage.

Definition 2.4.2. (EPIC) A mechanism (a^∗, χ, t) is said to be ex post incentive compatible if the following conditions are satisfied:

Vi(χ(a^∗(θ))), θ) − ti(a^∗(θ)) ≥ Vi(χ(ai, a^∗−i(θ⁻i)), θ) − ti(ai, a^∗−i(θ⁻i)), ∀ai ∈ A_i, ∀i ∈ N (2.4.2) Clearly, the outcomeχ(a^∗(θ)) is an ex post Nash equilibrium where every agent maxi-mizes her ex post utility for all possible realizations of other agents’ private information when all other agents play their equilibrium strategies as well. One can also conclude that if EPIC is satisfied, then IIC must be satisfied as well. In other words, satisfying EPIC is a sufficient condition for satisfying IIC.

Definition 2.4.3. (DSIC) A mechanism (a^∗, χ, t) is said to be dominant strategy incentive compatible if the following conditions are satisfied:

Vi(χ(a^∗_i(θi), a⁻i)), θ) − ti(a^∗_i(θi), a⁻i) ≥ Vi(χ(ai)), θ) − ti(a), ∀ai ∈A^t_i, ∀i ∈ N^t (2.4.3) DSIC mechanism aligns with dominant strategies in noncooperative game with perfect information and is the strongest condition of all three. This time, any arbitrary agent i under a^∗maximizes her ex post utility for all possible realizations of other agents’ privation information and all possible strategies others take.

2.4.2 Individual Rationality

Individual rationality describes the condition when agents are willing to join the game, rather than opt out. Suppose that agents who opt out the game will receive zero utility,

then it is rational for them to stay if they can obtain non-negative utility (either ex ante, interim, or ex post) by staying in the game. Similarly, individual rationality can be categorized into three classes.

Definition 2.4.4. (EAIR) A mechanism (a^∗, χ, t) is said to be ex ante individual rational if the following conditions are satisfied:

E[Vi(χ(a^∗(θ))), θ) − ti(a^∗(θ))] ≥ 0. (2.4.4) Definition 2.4.5. (IIR) A mechanism (a^∗, χ, t) is said to be interim individual rational if the following conditions are satisfied:

E[vi(χ(a^∗(θ))), θ) − ti(a^∗(θ))|θi] ≥ 0. (2.4.5) Definition 2.4.6. (EPIR) A mechanism (a^∗, χ, t) is said to be ex post individual rational if the following conditions are satisfied:

Vi(χ(a^∗(θ))), θ) − ti(a^∗(θ)) ≥ 0. (2.4.6)

2.4.3 Budget Balancing

The last constraint introduced is budget balancing, or revenue neutral. Under strict budget balancing, the sum of all transfers equals zero, meaning that the mechanism designer neither substitutes the mechanism, nor gains revenue from the mechanism. A relaxed condition is weakly budget balancing, where the sum of all transfers is non-negative. In this case, the mechanism designer gains profits from the mechanism, which is preferable in many applications. Indeed, many auctions are designed in the way that the expected sum of transfers is maximized. The following context provides the related notations of budget balancing.

Definition 2.4.7. (EABB) A mechanism (a^∗, χ, t) is said to be ex ante budget balance, if X

i

E[ti(a^∗(θ))] = 0 (2.4.7)

Definition 2.4.8. (EPWBB) A mechanism (a^∗, χ, t) is said to be ex post weakly budget balance, if

X

i

ti(a^∗(θ)) ≥ 0 (2.4.8)

Definition 2.4.9. (EPBB) A mechanism (a^∗, χ, t) is said to be ex post budget balance, if X

i

ti(a^∗(θ)) = 0 (2.4.9)

It should be mentioned that some literature also uses ex post budget balance loosely to refer ex post weakly budget balance. However, this thesis strictly distinguishes the two definitions.

CHAPTER 3

Estimating Energy-Saving Potentials: the

Planning Perspective