Strategyproofness

In game theory, an asymmetric game where players have private information is said to be strategyproof (SP) if it is a weakly-dominant strategy for every player to reveal his/her private information,^[1]^:244 i.e. you fare best or at least not worse by being truthful, regardless of what the others do.

SP is also called truthful or dominant-strategy-incentive-compatible (DSIC),^[1]^:415 to distinguish it from other kinds of incentive compatibility.

An SP game is not always immune to collusion, but its robust variants are; with group strategyproofness no group of people can collude to misreport their preferences in a way that makes every member better of, and with strong group strategyproofness no group of people can collude to misreport their preferences in a way that makes at least one member of the group better off without making any of the remaining members worse of.^[2]

Examples

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

Typical examples of a 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.

Notation

There is a set $X$ of possible outcomes.

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

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

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

The vector of all value-functions is denoted by $v$ .

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

A mechanism is a pair of functions:

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

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})

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

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

Outcome(v_{i},v_{-i})=Outcome(v_{i}',v_{-i})

then:

Payment_{i}(v_{i},v_{-i})=Payment_{i}(v_{i}',v_{-i})

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

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

Outcome(v_{i},v_{-i})=x

then:

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

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

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 $Outcome(\cdot ,v_{-i})$ .

PROOF: If there is another outcome $x'=Outcome(v_{i}',v_{-i})$ such that $v_{i}(x')+Price_{i}(x',v_{-i})>v_{i}(x)+Price_{i}(x,v_{-i})$ , then an agent with valuation $v_{i}$ prefers to report $v_{i}'$ , 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 $i$ and valuations $v_{i},v_{i}',v_{-i}$ . Denote:

x:=Outcome(v_{i},v_{-i})

- the outcome when the agent acts truthfully.

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:

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

and the utility of the agent when playing truthfully is:

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

By property 2:

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

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

Outcome-function characterization

The actual goal of a mechanism is its $Outcome$ function; the payment function is just a tool to induce the players to be truthful. Hence, it is useful to know, given a certain outcome function, whether it can be implemented using a SP mechanism or not (this property is also called implementability). The Monotonicity (mechanism design) property is necessary, and often also sufficient.

Truthful mechanisms in single-parameter domains

A single-parameter domain is a game in which each player i gets a certain positive value v_i for "winning" and a value 0 for "losing". A simple example is a single-item auction, in which v_i is the value that player 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

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

Truthfulness with-high-probability

For every constant $\epsilon >0$ , a randomized mechanism is called truthful with probability $1-\epsilon$ if for every agent and for every vector of bids, the probability that the agent benefits by bidding non-truthfully is at most $\epsilon$ , where the probability is taken over the randomness of the mechanism.^[1]^:349

If the constant $\epsilon$ goes to 0 when the number of bidders grows, then the mechanism is called truthful with high probability. This notion is weaker than full truthfulness, but it is still useful in some cases; see e.g. consensus estimate.

False-name-proofness

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.^[3]

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.

References

1 2 3 4 5 Vazirani, Vijay V.; Nisan, Noam; Roughgarden, Tim; Tardos, Éva (2007). Algorithmic Game Theory (PDF). Cambridge, UK: Cambridge University Press. ISBN 0-521-87282-0.
↑ "Group Strategy-proofness And Social Choice Between Two Alternatives" (PDF).
↑ Yokoo, M.; Sakurai, Y.; Matsubara, S. (2004). "The effect of false-name bids in combinatorial auctions: New fraud in internet auctions". Games and Economic Behavior. 46: 174. doi:10.1016/S0899-8256(03)00045-9.

Topics in game theory
Definitions	Normal-form game Extensive-form game Escalation of commitment Graphical game Cooperative game Succinct game Information set Hierarchy of beliefs Preference
Equilibrium concepts	Nash equilibrium Subgame perfection Mertens-stable equilibrium Bayesian Nash equilibrium Perfect Bayesian equilibrium Trembling hand Proper equilibrium Epsilon-equilibrium Correlated equilibrium Sequential equilibrium Quasi-perfect equilibrium Evolutionarily stable strategy Risk dominance Core Shapley value Pareto efficiency Gibbs equilibrium Quantal response equilibrium Self-confirming equilibrium Strong Nash equilibrium Markov perfect equilibrium
Strategies	Dominant strategies Pure strategy Mixed strategy Tit for tat Grim trigger Collusion Backward induction Forward induction Markov strategy
Classes of games	Symmetric game Perfect information Simultaneous game Sequential game Repeated game Signaling game Screening game Cheap talk Zero-sum game Mechanism design Bargaining problem Stochastic game n-player game Large Poisson game Nontransitive game Global game Strictly determined game Potential game
Games	Prisoner's dilemma Optional prisoner's dilemma Traveler's dilemma Coordination game Chicken Centipede game Volunteer's dilemma Dollar auction Battle of the sexes Stag hunt Matching pennies Ultimatum game Rock–paper–scissors Pirate game Dictator game Public goods game Blotto game War of attrition El Farol Bar problem Fair division Fair cake-cutting Cournot game Deadlock Diner's dilemma Guess 2/3 of the average Kuhn poker Nash bargaining game Prisoners and hats puzzle Trust game Princess and Monster game Rendezvous problem
Theorems	Minimax theorem Nash's theorem Purification theorem Folk theorem Revelation principle Arrow's impossibility theorem
Key figures	Albert W. Tucker Amos Tversky Ariel Rubinstein Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin Jean Tirole Jean-François Mertens John Harsanyi John Maynard Smith Antoine Augustin Cournot John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Axelrod Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Thomas Schelling William Vickrey
See also	All-pay auction Alpha–beta pruning Bertrand paradox Bounded rationality Combinatorial game theory Confrontation analysis Coopetition List of game theorists List of games in game theory No-win situation Topological game Tragedy of the commons Tyranny of small decisions

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.