Strategyproofness

Typical examples of SP mechanisms are majority voting between two alternatives, second-price auction, and any VCG mechanism.

Typical examples of mechanisms that are not SP are plurality voting between three or more alternatives, and first-price auction.

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.

There is a set ${\displaystyle X}$ of possible outcomes.

There are ${\displaystyle n}$ agents which have different valuations for each outcome. The valuation of agent ${\displaystyle i}$ is represented as a function:

${\displaystyle v_{i}:X\longrightarrow R_{+}}$

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 ${\displaystyle x}$ and in addition the agent receives a payment ${\displaystyle p_{i}}$ (positive or negative), then the total utility of agent ${\displaystyle i}$ is:

${\displaystyle u_{i}:=v_{i}(x)+p_{i}}$

The vector of all value-functions is denoted by ${\displaystyle v}$.

For every agent ${\displaystyle i}$, the vector of all value-functions of the other agents is denoted by ${\displaystyle v_{-i}}$. So ${\displaystyle v\equiv (v_{i},v_{-i})}$.

A mechanism is a pair of functions:

• An ${\displaystyle Outcome}$ function, that takes as input the value-vector ${\displaystyle v}$ and returns an outcome ${\displaystyle x\in X}$ (it is also called a social choice function);
• A ${\displaystyle Payment}$ function, that takes as input the value-vector ${\displaystyle v}$ and returns a vector of payments, ${\displaystyle (p_{1},\dots ,p_{n})}$, 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 ${\displaystyle i}$ and for every value-vector of the other players ${\displaystyle v_{-i}}$:

${\displaystyle v_{i}(Outcome(v_{i},v_{-i}))+Payment_{i}(v_{i},v_{-i})\geq v_{i}(Outcome(v_{i}',v_{-i}))+Payment_{i}(v_{i}',v_{-i})}$

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 is SP, then it must satisfy the following two conditions, for every agent ${\displaystyle i}$:[1]^: 226

1. The payment to agent ${\displaystyle i}$ is a function of the chosen outcome and of the valuations of the other agents ${\displaystyle v_{-i}}$ - but not a direct function of the agent's own valuation ${\displaystyle v_{i}}$. Formally, there exists a price function ${\displaystyle Price_{i}}$, that takes as input an outcome ${\displaystyle x\in X}$ and a valuation vector for the other agents ${\displaystyle v_{-i}}$, and returns the payment for agent ${\displaystyle i}$, such that for every ${\displaystyle v_{i},v_{i}',v_{-i}}$, if:

${\displaystyle Outcome(v_{i},v_{-i})=Outcome(v_{i}',v_{-i})}$

then:

${\displaystyle Payment_{i}(v_{i},v_{-i})=Payment_{i}(v_{i}',v_{-i})}$

PROOF: If ${\displaystyle Payment_{i}(v_{i},v_{-i})>Payment_{i}(v_{i}',v_{-i})}$ then an agent with valuation ${\displaystyle v_{i}'}$ prefers to report ${\displaystyle v_{i}}$, since it gives him the same outcome and a larger payment; similarly, if ${\displaystyle Payment_{i}(v_{i},v_{-i})<Payment_{i}(v_{i}',v_{-i})}$ then an agent with valuation ${\displaystyle v_{i}}$ prefers to report ${\displaystyle v_{i}'}$.

As a corollary, there exists a "price-tag" function, ${\displaystyle Price_{i}}$, that takes as input an outcome ${\displaystyle x\in X}$ and a valuation vector for the other agents ${\displaystyle v_{-i}}$, and returns the payment for agent ${\displaystyle i}$ For every ${\displaystyle v_{i},v_{-i}}$, if:

${\displaystyle Outcome(v_{i},v_{-i})=x}$

then:

${\displaystyle Payment_{i}(v_{i},v_{-i})=Price_{i}(x,v_{-i})}$

2. The selected outcome is optimal for agent ${\displaystyle i}$, given the other agents' valuations. Formally:

${\displaystyle Outcome(v_{i},v_{-i})\in \arg \max _{x}[v_{i}(x)+Price_{i}(x,v_{-i})]}$

where the maximization is over all outcomes in the range of ${\displaystyle Outcome(\cdot ,v_{-i})}$.

PROOF: If there is another outcome ${\displaystyle x'=Outcome(v_{i}',v_{-i})}$ such that ${\displaystyle v_{i}(x')+Price_{i}(x',v_{-i})>v_{i}(x)+Price_{i}(x,v_{-i})}$, then an agent with valuation ${\displaystyle v_{i}}$ prefers to report ${\displaystyle v_{i}'}$, since it gives him a larger total utility.

Conditions 1 and 2 are not only necessary but also sufficient: any mechanism that satisfies conditions 1 and 2 is SP.

PROOF: Fix an agent ${\displaystyle i}$ and valuations ${\displaystyle v_{i},v_{i}',v_{-i}}$. Denote:

${\displaystyle x:=Outcome(v_{i},v_{-i})}$ - the outcome when the agent acts truthfully.

${\displaystyle x':=Outcome(v_{i}',v_{-i})}$ - the outcome when the agent acts untruthfully.

By property 1, the utility of the agent when playing truthfully is:

${\displaystyle u_{i}(v_{i})=v_{i}(x)+Price_{i}(x,v_{-i})}$

and the utility of the agent when playing untruthfully is:

${\displaystyle u_{i}(v_{i}')=v_{i}(x')+Price_{i}(x',v_{-i})}$

By property 2:

${\displaystyle u_{i}(v_{i})\geq u_{i}(v_{i}')}$

so it is a dominant strategy for the agent to act truthfully.

A single-parameter domain is a game in which each player ${\displaystyle i}$ gets a certain positive value ${\displaystyle v_{i}}$ for "winning" and a value 0 for "losing". A simple example is a single-item auction, in which ${\displaystyle v_{i}}$ is the value that player ${\displaystyle i}$ assigns to the item.

For this setting, it is easy to characterize truthful mechanisms. Begin with some definitions.

A mechanism is called normalized if every losing bid pays 0.

A mechanism is called monotone if, when a player raises his bid, his chances of winning (weakly) increase.

For a monotone mechanism, for every player i and every combination of bids of the other players, there is a critical value in which the player switches from losing to winning.

A normalized mechanism on a single-parameter domain is truthful iff the following two conditions hold:[1]^{: 229–230}

1. The assignment function is monotone in each of the bids, and:
2. Every winning bid pays the critical value.

There are various ways to extend the notion of truthfulness to randomized mechanisms. They are, from strongest to weakest:[3]^: 6–8

• 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.[3]^: Thm.3.4

A new type of fraud that has become common with the abundance of internet-based auctions is false-name bids – bids submitted by a single bidder using multiple identifiers such as multiple e-mail addresses.

False-name-proofness means that there is no incentive for any of the players to issue false-name-bids. This is a stronger notion than strategyproofness. In particular, the Vickrey–Clarke–Groves (VCG) auction is not false-name-proof.[4]

False-name-proofness is importantly different from group strategyproofness because it assumes that an individual alone can simulate certain behaviours that would normally require the collusive coordination of multiple individuals.

• Incentive compatibility
• Individual rationality - means that a player cannot lose by playing the game (i.e. a player has no incentive to avoid playing the game).

• Parkes, David C. (2004), On Learnable Mechanism Design, in: Tumer, Kagan and David Wolpert (Eds.): Collectives and the Design of Complex Systems, New York u.a.O., pp. 107–133.
• On Asymptotic Strategy-Proofness of Classical Social Choice Rules An article by Arkadii Slinko about strategy-proofness in voting systems.