The overall objective of the prosumer coordination algorithm is to minimise the total system costs (see section 4.4.3). This can be rephrased within the field of social choice, which investigates how to combine individual agent preferences, in order to produce a collective decision [Shoham and Leyton- Brown, 2009]. Preferences are combined by defining asocial choice function
that maps agent preferences to an overall outcome.
For our power systems problem, prosumers and other participants are the agents, their preferences are over different patterns of electricity con- sumption and the collective outcome is the allocation of power for each agent. The social choice function for this problem returns an outcome that minimises the total system costs.
Mechanism design is a sub-field of game theory and economics that is related to social choice. Amechanism defines the rules around how agents interact and how decisions are made in a strategic setting where agents can lie about their preferences [Nisan et al., 2007, Shoham and Leyton-Brown, 2009]. The goal of a mechanism is to produce the same outcome as a social choice function, but in a strategic setting.
Auctions are an example of a class of mechanism, that can be used for allocating a finite resource to potential buyers. In this setting, the social choice function might be to allocate an item to the agent that values it most. For a particular mechanism, game theory can be applied to find the out- come. When agents know the true preferences of all other agents, it is called a complete information game1. More commonly, if agents only have beliefs
1
5.2. MECHANISM DESIGN 85 about the preferences of others, this is called an incomplete information game. A mechanism successfully implements a social choice function if it creates a game that has dominant strategies or Bayes-Nash equilibria that produce the same outcome as the social choice function applied to the true agent preferences [Shoham and Leyton-Brown, 2009, definitions 10.2.3-4].
5.2.1 Incentive Compatibility
An agent manipulates a mechanism when it gains an advantage by being untruthful (see Nisan et al. [2007, definition 9.4]), which can prevent the mechanism from achieving the social outcome. If it is not possible for agents to manipulate a mechanism (their best strategy is to be truthful) then it is
incentive compatible. More precisely, a mechanism is:
• dominant-strategy incentive compatible if being truthful is a weakly- dominant strategy for agents; or
• Bayes-Nash incentive compatible if there is a Bayes-Nash equilibrium where all agents act truthfully.
Unfortunately, there are strong theoretical results that show incentive com- patibility to be incompatible with other desirable properties.
One of the leading results is the Gibbard-Satterthwaite theorem. It states that for a social choice problem with three or more outcomes, any incentive compatible social choice function that does not preclude an out- come from winning must be a dictatorship [Nisan et al., 2007, theorem 9.8]. A dictatorship, where one agent gets to decide the outcome for the whole system, is clearly not a desirable property.
This suggests that it is futile to design any useful incentive compatible social choice function; however, this result is for the case where agents are allowed to specify arbitrary preferences. By restricting the class of pref- erences, it turns out that one can develop useful mechanisms that achieve incentive compatibility.
5.2.2 Vickrey-Clarke-Groves Mechanisms
One way of restricting the class of preferences is to work withutilities and to introduce monetary payments. An agent’s utility function maps each out- come to a real value, where the larger the value the greater their preference. This utility function also takes as input any money paid to, or paid by, the agent. A common assumption is that the money appears as a linear term in the utility function, making it a quasilinear utility function:
u(o, m) =m+v(o) (5.1)
not have access to the agents’ true preferences, otherwise it could trivially just apply the social choice function to determine the outcome.
where o is an outcome, m is money (positive when paid to agent), u is a utility function and v is a valuation function. Here we have made the assumption that the agent is risk neutral (otherwise the money term appears as a monotonically increasing function) [Shoham and Leyton-Brown, 2009, definition 10.3.1]. We also assume that the property of transferable utility holds, allowing utility to be transferred between agents through payments.
When money is involved, mechanisms have a rule for choosing an out- come given the preferences, and a rule for allocating payments to agents. This provides mechanisms with greater flexibility, where payments can be used to induce incentive compatibility.
A commonly used social choice function in this setting is the maximisa- tion of social welfare, where social welfare for a particular outcome is the sum of all agent valuations: P
i∈Avi(o). This is equivalent to our desire to
minimise total system costs for the prosumer coordination problem.
Vickrey-Clarke-Groves (VCG) mechanisms are an important family of mechanisms, that break free of the Gibbard-Satterthwaite result within this new restricted preference setting. They maximise social welfare and are dominant-strategy incentive compatible. They achieve this because the pay- ments are designed in such a way that each agent’s utility becomes dependent on the social welfare.
Many of the mechanisms in the VCG family are impractical because they are notbudget balanced (the net sum of payments in the mechanism is non-zero), requiring large amounts of funds to be externally sourced or sunk. One common VCG mechanism (often referred to as the VCG mechanism) uses the Clarke pivot rule to fully define the payments. Under some, often mild, additional conditions, this rule achieves the desirable properties of weak budget balance and individual rationality.
A mechanism is weakly budget balanced when the net sum of payments made to mechanism is non-negative. This means that only an external sink of funds is required, which is much easier to achieve than an external source (in the worst case the money can be burnt). A mechanism is individually rational if all agents receive a positive utility. The idea is that if this were not the case, then certain agents would not participate in the mechanism (assuming they have a utility of zero for not participating).
The VCG mechanism has some great properties as we have discussed, but it does come with its negatives [Shoham and Leyton-Brown, 2009, section 10.4.5]. It requires agents to fully reveal their preferences to the mechanism, which has severe privacy implications. Also, it can be computationally in- tractable to calculate agent payments (largely due to the Clarke pivot rule) for many real-sized problems.
Another problem is that the weak budget balance property only holds in certain domains. For example, it no longer holds in exchange settings where there are multiple buyers and sellers (as is the case in our prosumer coordination problem). Also, while VCG is incentive compatible in a fully
5.3. POWER BALANCING PROBLEM 87