Search Results for “strategyproofness”

Source: 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.: 244 An SP mechanism is also called dominant-strategy-incentive-compatible (DSIC),: 415 to distinguish it from other kinds of incentive compatibility.
An SP mechanism is immune to manipulations by individual players (but not by coalitions). In contrast, in a group strategyproof mechanism, no group of people can collude to misreport their preferences in a way that makes every member better off. In a strong group strategyproof mechanism, 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 off.

Examples

= SP in network routing

Formal definitions

Characterization

then an agent with valuation

v

i

{\displaystyle v_{i}}

prefers to report

v

i

′

{\displaystyle v_{i}'}

.
As a corollary, there exists a "price-tag" function,

P
r
i
c

e

i

{\displaystyle Price_{i}}

, that takes as input an outcome

x
∈
X

{\displaystyle x\in X}

and a valuation vector for the other agents

v

−
i

{\displaystyle v_{-i}}

, and returns the payment for agent

i

{\displaystyle i}

For every

v

i

,

v

−
i

{\displaystyle v_{i},v_{-i}}

, if:

O
u
t
c
o
m
e
(

v

i

,

v

−
i

)
=
x

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

then:

P
a
y
m
e
n

t

i

(

v

i

,

v

−
i

)
=
P
r
i
c

e

i

(
x
,

v

−
i

)

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

2. The selected outcome is optimal for agent

i

{\displaystyle i}

, given the other agents' valuations. Formally:

O
u
t
c
o
m
e
(

v

i

,

v

−
i

)
∈
arg
⁡

max

x

[

v

i

(
x
)
+
P
r
i
c

e

i

(
x
,

v

−
i

)
]

{\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

O
u
t
c
o
m
e
(
⋅
,

v

−
i

)

{\displaystyle Outcome(\cdot ,v_{-i})}

.
PROOF: If there is another outcome

x
′

=
O
u
t
c
o
m
e
(

v

i

′

,

v

−
i

)

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

such that

v

i

(

x
′

)
+
P
r
i
c

e

i

(

x
′

,

v

−
i

)
>

v

i

(
x
)
+
P
r
i
c

e

i

(
x
,

v

−
i

)

{\displaystyle v_{i}(x')+Price_{i}(x',v_{-i})>v_{i}(x)+Price_{i}(x,v_{-i})}

, then an agent with valuation

v

i

{\displaystyle v_{i}}

prefers to report

v

i

′

{\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

i

{\displaystyle i}

and valuations

v

i

,

v

i

′

,

v

−
i

{\displaystyle v_{i},v_{i}',v_{-i}}

. Denote:

x
:=
O
u
t
c
o
m
e
(

v

i

,

v

−
i

)

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

- the outcome when the agent acts truthfully.

x
′

:=
O
u
t
c
o
m
e
(

v

i

′

,

v

−
i

)

{\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:

u

i

(

v

i

)
=

v

i

(
x
)
+
P
r
i
c

e

i

(
x
,

v

−
i

)

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

and the utility of the agent when playing untruthfully is:

u

i

(

v

i

′

)
=

v

i

(

x
′

)
+
P
r
i
c

e

i

(

x
′

,

v

−
i

)

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

By property 2:

u

i

(

v

i

)
≥

u

i

(

v

i

′

)

{\displaystyle 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

O
u
t
c
o
m
e

{\displaystyle 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 property is necessary for strategyproofness.

Truthful mechanisms in single-parameter domains

A single-parameter domain is a game in which each player

i

{\displaystyle i}

gets a certain positive value

v

i

{\displaystyle v_{i}}

for "winning" and a value 0 for "losing". A simple example is a single-item auction, in which

v

i

{\displaystyle v_{i}}

is the value that player

i

{\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 if the following two conditions hold:: 229–230

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

Truthfulness of randomized mechanisms

There are various ways to extend the notion of truthfulness to randomized mechanisms. They are, from strongest to weakest:: 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.: Thm.3.4

= Truthfulness with high probability

=
For every constant

ϵ
>
0

{\displaystyle \epsilon >0}

, a randomized mechanism is called truthful with probability

1
−
ϵ

{\displaystyle 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

ϵ

{\displaystyle \epsilon }

, where the probability is taken over the randomness of the mechanism.: 349
If the constant

ϵ

{\displaystyle \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.
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.