Strategyproofness
Strategyproofness
In mechanism-design, a strategyproof (SP) mechanism is a game-form in which each player has a weakly-dominant strategy, so that no player can gain by "spying" over the other players to know what they are going to play. When the players have private information (e.g. their type or their value to some item), and the strategy space of each player consists of the possible information values (e.g. possible types or values), a truthful mechanism is a game in which revealing the true information is a weakly-dominant strategy for each player. An SP mechanism is also called dominant-strategy-incentive-compatible (DSIC),
Examples Typical examples of SP mechanisms are:
- a majority vote between two alternatives;
- a second-price auction when participants have quasilinear utility;
- a VCG mechanism when participants have quasilinear utility
Typical examples of mechanisms that are not SP are:
- any deterministic non-dictatorial election between three or more alternatives;
- a first-price auction
SP in network routing SP is also applicable in network routing. Consider a network as a graph where each edge (i.e. link) has an associated cost of transmission, privately known to the owner of the link. The owner of a link wishes to be compensated for relaying messages. As the sender of a message on the network, one wants to find the least cost path. There are efficient methods for doing so, even in large networks. However, there is one problem: the costs for each link are unknown. A naive approach would be to ask the owner of each link the cost, use these declared costs to find the least cost path, and pay all links on the path their declared costs. However, it can be shown that this payment scheme is not SP, that is, the owners of some links can benefit by lying about the cost. We may end up paying far more than the actual cost. It can be shown that given certain assumptions about the network and the players (owners of links), a variant of the VCG mechanism is SP.
Formal definitions There is a set <math>X</math> of possible outcomes.
There are <math>n</math> agents which have different valuations for each outcome. The valuation of agent <math>i</math> is represented as a function:
: <math>v_i : X \longrightarrow R_+</math>
which expresses the value it has for each alternative, in monetary terms.
It is assumed that the agents have Quasilinear utility functions; this means that, if the outcome is <math>x</math> and in addition the agent receives a payment <math>p_i</math> (positive or negative), then the total utility of agent <math>i</math> is:
: <math>u_i := v_i(x) + p_i</math>
The vector of all value-functions is denoted by <math>v</math>.
For every agent <math>i</math>, the vector of all value-functions of the other agents is denoted by <math>v_{-i}</math>. So <math>v \equiv (v_i,v_{-i})</math>.
A mechanism is a pair of functions: An <math>Outcome</math> function, that takes as input the value-vector <math>v</math> and returns an outcome <math>x\in X</math> (it is also called a social choice function); A <math>Payment</math> function, that takes as input the value-vector <math>v</math> and returns a vector of payments, <math>(p_1,\dots,p_n)</math>, determining how much each player should receive (a negative payment means that the player should pay a positive amount).
A mechanism is called strategyproof if, for every player <math>i</math> and for every value-vector of the other players <math>v_{-i}</math>: : <math>v_i(Outcome(v_i,v_{-i})) + Payment_i(v_i,v_{-i}) \geq v_i(Outcome(v_i',v_{-i})) + Payment_i(v_i',v_{-i})</math>
Characterization It is helpful to have simple conditions for checking whether a given mechanism is SP or not. This subsection shows two simple conditions that are both necessary and sufficient.
If a mechanism with monetary transfers is SP, then it must satisfy the following two conditions, for every agent <math>i</math>:
- Universal truthfulness: for each randomization of the algorithm, the resulting mechanism is truthful. In other words: a universally-truthful mechanism is a randomization over deterministic truthful mechanisms, where the weights may be input-dependent.
- Strong stochastic-dominance truthfulness (strong-SD-truthfulness): The vector of probabilities that an agent receives by being truthful has first-order stochastic dominance over the vector of probabilities he gets by misreporting. That is: the probability of getting the top priority is at least as high AND the probability of getting one of the two top priorities is at least as high AND ... the probability of getting one of the m top priorities is at least as high.
- Lexicographic truthfulness (lex-truthfulness): The vector of probabilities that an agent receives by being truthful has lexicographic dominance over the vector of probabilities he gets by misreporting. That is: the probability of getting the top priority is higher OR (the probability of getting the top priority is equal and the probability of getting one of the two top priorities is higher) OR ... (the probability of getting the first m-1 priorities priority is equal and the probability of getting one of the m top priorities is higher) OR (all probabilities are equal).
- Weak stochastic-dominance truthfulness (weak-SD-truthfulness): The vector of probabilities that an agent receives by being truthful is not first-order-stochastically-dominated by the vector of probabilities he gets by misreporting.
Universal implies strong-SD implies Lex implies weak-SD, and all implications are strict.
False-name-proofness is importantly different from group strategyproofness because it assumes that an individual alone can simulate certain behaviors that normally require the collusive coordination of multiple individuals.
Obvious strategyproofness Obvious strategyproofness (OSP) is a strengthening of strategyproofness that captures a robustness of strategyproofness to cognitively-limited agents.
The sealed-bid second-price auction is strategyproof but not obviously strategyproof because bidders have to trust that their bids remain sealed. In contrast, the ascending clock auction is obviously strategyproof, * Top trading cycles