Algorithmic Game Theory(because a game is nice when it’s not too long!)

Guido Proietti

Dipartimento di Informatica, Università degli Studi dell’Aquila&

Istituto di Analisi dei Sistemi ed Informatica – CNR Roma

Roadmap

Nash Equilibria (NE) Does a NE always exist? Can a NE be feasibly computed, once it exists? Which is the “quality” of a NE? How long does it take to converge to a NE?

Algorithmic Mechanism Design Which social goals can be (efficiently)

implemented in a non-cooperative selfish distributed system?

VCG-mechanisms and one-parameter mechanisms

Mechanism design for some basic network design problems

FIRST PART:Nash equilibria

Two Research Traditions

Theory of Algorithms: computational issues What can be feasibly computed? How much does it take to compute a solution? Which is the quality of a computed solution? Centralized or distributed computational models

Game Theory: interaction between self-interested individuals What is the outcome of the interaction? Which social goals are compatible with

selfishness?

Different Assumptions

Theory of Algorithms (in DCMs): Processors are obedient, faulty, or

adversarial Large systems, limited comp. resources

Game Theory: Players are strategic (selfish) Small systems, unlimited comp.

resources

The Internet World

Agents often autonomous (users) Users have their own individual goals Network components owned by providers

Often involve “Internet” scales Massive systems Limited communication/computational

resources

Both strategic and computational issues!

Fundamental Questions

What are the computational aspects of a

game?

What does it mean to design an algorithm

for a strategic distributed system?Theory of Algorithms

Game Theory

AlgorihmicGame Theory

+=

Game Theory

Given a game, predict the outcome by analyzing the individual behavior of the players (agents)

Game: N players Rules of encounter: Who should act

when, and what are the possible actions

Outcomes of the game

Normal Form Games

N rational and non-cooperative players Si =Strategy set of player i The strategy combination (s1, s2, …, sN) gives

payoff (p1, p2, …, pN) to the N players All the above information is known to all the

players and it is common knowledge Simultaneous move: each player i chooses a

strategy siSi (nobody can observe others’ move)

Equilibrium

An equilibrium s*= (s1*, s2

*, …, sN*) is a

strategy combination consisting of a best strategy for each of the N players in the game

What is a best strategy? depends on the game…informally, it is a strategy that a players selects in trying to maximize his individual payoff, knowing that other players are also doing the same

Dominant Strategy Equilibrium: Prisoner’s Dilemma

Prisoner I

Prisoner II

Don’t Implicate

Implicate

Don’t Implicate

-1, -1 -6, 0

Implicate 0, -6 -5, -5

StrategySet

Strategy Set

Payoffs

Prisoner I’s decision

Prisoner I’s decision: If II chooses Don’t Implicate then it is best to Implicate If II chooses Implicate then it is best to Implicate It is best to Implicate for I, regardless of what II does:

Dominant Strategy

Prisoner I

Prisoner II

Don’t Implicate

Implicate

Don’t Implicate

-1, -1 -6, 0

Implicate 0, -6 -5, -5

Prisoner II’s decision

Prisoner II’s decision: If I chooses Don’t Implicate then it is best to Implicate If I chooses Implicate then it is best to Implicate It is best to Implicate for II, regardless of what I does:

Dominant Strategy

Prisoner I

Prisoner II

Don’t Implicate

Implicate

Don’t Implicate

-1, -1 -6, 0

Implicate 0, -6 -5, -5

Hence…

It is best for both to implicate regardless of what the other one does Implicate is a Dominant Strategy for both (Implicate, Implicate) becomes the Dominant Strategy Equilibrium Note: If they might collude, then it’s beneficial for both to Not

Implicate, but it’s not an equilibrium as both have incentive to deviate

Prisoner I

Prisoner II

Don’t Implicate

Implicate

Don’t Implicate

-1, -1 -6, 0

Implicate 0, -6 -5, -5

Dominant Strategy Equilibrium

Dominant Strategy Equilibrium: is a strategy combination s*= (s1

*, s2*, …, sN

*), such that si

* is a dominant strategy for each i, namely, for each s= (s1, s2, …, si , …, sN):

pi (s1, s2, …, si

*, …, sN) ≥ pi (s1, s2, …, si, …, sN)

Dominant Strategy is the best response to any strategy of other players

It is good for agent as it needs not to deliberate about other agents’ strategies

Not all games have a dominant strategy equilibrium

A Beautiful Mind: Nash Equilibrium

Nash Equilibrium: is a strategy combination s*= (s1

*, s2*, …, sN

*) such that for each i, si* is a

best response to (s1*, …,si-1

*,si+1*,…, sN

*), namely, for any possible alternative strategy si

pi (s*) ≥ pi

(s1*, s2

*, …, si, …, sN*)

Note: We are playing simultaneous games, and so nobody knows a priori the choice of other agents

Nash Equilibrium

In a NE no agent can unilaterally deviate from its strategy given others’ strategies as fixed

There may be no, one or many NE, depending on the game

Agent has to deliberate about the strategies of the other agents

If the game is played repeatedly and players converge to a solution, then it has to be a NE

Dominant Strategy Equilibrium Nash Equilibrium (but the converse is not true)

Nash Equilibrium: The Battle of the Sexes (coordination game)

(Stadium,Stadium) is a NE: Best responses to each other

(Cinema, Cinema) is a NE: Best responses to each other

but they are not Dominant Strategy Equilibria … are we really sure they will eventually go out together????

Man

Woman

Stadium Cinema

Stadium 2, 1 0, 0

Cinema 0, 0 1, 2

A conflictual game: Head or Tail

Player I (row) prefers to do what Player II does, while Player II prefer to do the opposite of what Player I does!

In any configuration, one of the players prefers to change his strategy, and so on and so forth…thus, there are no NE!

Player I

Player II

Head Tail

Head 1,-1 -1,1

Tail -1,1 1,-1

Three big computational issues

1. Finding a NE, once it does exist2. Establishing the quality of a NE, as

compared to a cooperative system, i.e., a system in which agents can cooperate (recall the Prisoner’s Dilemma)

3. In a repeated game, establishing whether and in how many steps the system will eventually converge to a NE (recall the Battle of the Sex)

On the existence of a NE

Theorem (Nash, 1951): Any game with a finite set of players and finite set of strategies has a NE of mixed strategies.

Mixed strategies: each player independently selects his strategy by using a probability distribution over his set of possible strategies

Head or Tail game: if each player sets p(Head)=p(Tail)=1/2, then the expected payoff of each player is 0, and this is a NE, since no player can improve on this by choosing a different randomization!

On the computability of a NE

But how do we select this probability distribution?It looks like a problem in the continuous……but it’s not, actually! It can be shown that such a distribution can be found by checking for all the (exponentially large) possible combinations for each player of the underlying pure strategies!

And why should we be interested on that?Because “If your laptop cannot find a NE, then the market probably cannot either”

Is finding a NE NP-hard?

W.l.o.g., we restrict ourself to 2-player games: The problem can be solved by a simplex-like technique called the Lemke–Howson algorithm, which however is exponential in the worst case

Reminder: a problem is NP-hard if one can reduce any NP-complete problem ’ to it:

“yes”-instance of ’ → “yes”-instance of “no”-instance of ’ → “no”-instance of

But each instance of 2-NASH is a “yes”-instance! (since every game has a NE)

if 2-NASH is NP-hard then NP = coNP (hard to believe!)

The complexity class PPAD

Definition (Papadimitriou, 1994): roughly speaking, PPAD (Polynomial Parity Argument – Directed case) is the class of all problems whose solution space can be set up as the set of all sinks in a suitable directed graph (generated by the input instance), having an exponential number of vertices in the size of the input, though.

Remark: It could very well be that PPAD=PNP……but several PPAD-complete problems are resisting for decades to poly-time attacks (e.g., finding Brouwer fixed points)

2-NASH is PPAD-complete!

3D-BROUWER is PPAD-complete (Papadimitriou, JCSS’94)

4-NASH is PPAD-complete (Daskalakis, Goldberg, and Papadimitriou, STOC’06)

3-NASH is PPAD-complete (Daskalakis & Papadimitriou, ECCC’05, Chen & Deng, ECCC’05)

2-NASH is PPAD-complete !!! (Chen & Deng, FOCS’06)

On the quality of a NE

How inefficient is a NE in comparison to an idealized situation in which the players would strive to collaborate selflessly with the common goal of maximazing the social welfare?

Recall: in the Prisoner’s Dilemma game, the DSE NE means a total of 10 years in jail for the players. However, if they would not implicate reciprocally, then they would stay a total of only 2 years in jail!

The price of anarchy

Definition (Koutsopias & Papadimitriou, 1999): Given a game G and a social-choice minimization (resp., maximization) function f (i.e., the sum of all players’ payoffs), let S be the set of NE, and let OPT be the outcome of G optimizing f. Then, the Price of Anarchy (PoA) of G w.r.t. f is:

Example: in the PD game, G(f)=-10/-2=5

Ssf

sup)(G

(OPT)

)(inf.,resp

(OPT)

)(

f

sf

f

sfSs

Internet components are made up of heterogeneous nodes and links, and the network architecture is open-based and dynamic

Internet users behave selfishly: they generate traffic, and their only goal is to download/upload data as fast as possible!

But the more a link is used, the more is slower, and there is no central authority “optimizing” the data flow…

So, why does Internet eventually work is such a jungle???

A case study: selfish routing on Internet

Latency is fixed

Latency depends on the congestion (x is the

fraction of flow using the edge)

s t

1)( x

Example: Pigou’s game (network congestion game)

What is the NE of this game? Trivial: all the fraction of flow tends to travel on the upper edge the cost of the flow is 1·1 +0·1 =1

What is the PoA of this NE? The optimal solution is the minimum of f(x)=x·x +(1-x)·1 f ’(x)=2x-1 OPT=1/2 f(OPT)=1/2·1/2+(1-1/2)·1=0.75 G(f) = 1/0.75 = 4/3

xx )(

Flows and NE

Assume now we are given a directed graph G = (V,E) and a set of source–sink pairs si,ti V between which selfish users want to push a certain amount of flow. Then, a flow is at Nash equilibrium (or is a Nash flow) if no agent can improve its latency by changing its path

Theorem (Beckmann et al., 1956): If edge latency functions are continuous and non-decreasing, and users control an infinitesimal amount of flow, then the Nash flow exists and is unique.

Flows and Price of Anarchy

Theorem 1: In a network with linear latency functions, the cost of a Nash flow is at most 4/3 times that of the minimum-latency flow.

Theorem 2: In a network with general latency functions, the cost of a Nash flow is at most n/2 times that of the minimum-latency flow.

(Roughgarden & Tardos, JACM’02)

A bad example for non-linear latencies

Assume i>>1

s t

xi

10

1 1-

A Nash flow (of cost 1) is arbitrarily more expensive than the optimal flow (of cost close to 0)

Convergence towards a NE(in pure strategies games)

Ok, we know that selfish routing is not so bad at its NE, but are we really sure this point of equilibrium will be eventually reached?

Convergence Time: number of moves made by the players to reach a NE from an initial arbitrary state

Question: Is the convergence time (polynomially) bounded in the number of players?

The potential function method

Roughly speaking, a potential function for a game is a real-valued function, defined on the set of possible outcomes of the game, such that the equilibria of the game are precisely the local optima of the potential function

Potential games: broad class of games admitting a potential function

Theorem: In any finite potential game, best response dynamics always converge to a NE of pure strategies.

Potential function and congestion games

How many steps are needed to reach a NE? It depends on the combinatorial structure of the players' strategy space

Definition (Matroid Congestion Games): A congestion game G is called MCG if: The strategy space of every player is the basis of a

matroid over the set of congested resources (recall that the size of this strategy space corresponds to the rank of the player's matroid);

The rank of the game, r(G), is defined to be the maximum matroid rank over all players.

Convergence in congestion games

Theorem (Achermann et al., FOCS'06): In a MCG G with n players and m resources, all best response improvement sequences have length O(n2m r(G)).

Example of a MCG: Load balancing Instead, in general, a network congestion game is

not a MCG Moreover, it is possible to show that there exist

instances for which the convergence time is exponential (unless finding a local optimum in any Polynomial Local Search (PLS class) problem can be done in polynomial time, against the common belief)

Still, Internet works quite fine!

And now…

SECOND PART:Algorithmic Mechanism Design(or, the art of convincing a capitalist to behave like a socialist )

Mechanism Design: the goal

Given: System comprising of self-interested, rational

agents Set of system-wide goals

Mechanism Design Does there exist a mechanism that can

implement the goals? Implementation of the goals depends on the

individual behavior of the agents

Mechanism Design: a picture

Agent 1

Agent n

Mechanism

p1

pn

tn

t

1r

1

r

n

Private “types” Reported types

Payments

Output

Each agent reports strategically to maximize its well-being…

…in response to a payment which is a function of the output!

Overview of the results

Algorithms Mechanisms

Centralized Decentralized (Non-cooperative) Network design problems

Mechanism Design

Games induced by mechanisms are different from games in standard form: Players hold independent private values The payoff matrix is a function of these types each player doesn’t really know about the

other players’ payoffs, but only about its one!

Games of Incomplete information Dominant Strategy Equilibrium is used

Mechanism Design Problem: Type of an Agent

N agents, and each agent has some private information called the type, tiTi, and performs a strategic action

We restrict ourself to direct revelation mechanisms, in which the action is reporting a type riTi (with possibly ri ti) Example: Auction Game

Each agent knows its cost for doing a job, but not the others’ one; the type of the agent is its cost

Ti= [0, +]: The agent’s cost may be any positive amount of

money ti= 80: Minimum amount of money the agent i is willing to be paid ri= 85: Exact amount of money the agent i bids to the system for

doing the job (not known to other agents)

Mechanism Design Problem: Output Specification

F is the set of feasible outputs Output Specification: For a given reported-type

configuration r=(r1, r2, …, rN), it specifies a valid outcome x(r)F which should optimize an objective function f(t) (the so-called social choice function) Example: Auction Game

F : Different winners of the auction f(t): mini (t1, t2, …, tN) (the lowest true cost) x(r): allocate to the bidder with lowest reported cost

Mechanism Design Problem: Valuation and Utility

If x is the outcome, then the valuation that agent i makes of x is given by a real valued function: vi(ti,x) Auction Game: If agent i wins the auction then its

valuation is equal to its actual cost for doing the job, otherwise it is 0

If pi is the payment given to the agent, then the utility of the outcome x is:

ui(ti,x) = pi - vi(ti,x) Auction Game: If agent’s cost for the job is 90,

and it gets the contract for 100 (i.e., it is paid 100), then its utility is 10

The Mechanism

A mechanism is a pair M=<x=g(r), p(x)> specifying: An algorithm g(r) which computes the

outcome x as a function of the reported types r

A payment scheme p as a function of the output x

Strategy-Proof Mechanisms

If truth telling is the dominant strategy in a mechanism then it is called Strategy-Proof Agents report their true types instead

of strategically manipulating it Utilitarian Problems: A problem is

utilitarian if its objective function is such that f(t) = i vi(ti,x) The Auction game is utilitarian

Vickrey-Clarke-Groves (VCG) Mechanisms

A VCG-mechanism is (the only) strategy-proof mechanism for utilitarian problems: Algorithm:

g(r)= arg maxxF i vi(ri,x) Payment function:

pi (x)= hi(r-i) - j≠i vj(rj,x)where hi(r-i) is an arbitrary function of the types of other

players What about non-utilitarian problems? We will see…

VCG-Mechanisms are Strategy-Proof

Proof (Intuitive sketch): Payment given to agent i

pi (x)= hi(r-i)-j≠i vj(rj,x)

and both the terms above are independent of the type, strategy and valuation of agent i

So it is best for agent i to report its true value. Strategic behavior does not lead to a beneficial outcome.

■

Clarke Mechanisms

This is a special VCG-mechanism (known as Clarke mechanism) in which

hi(r-i)=j≠i vj(rj,x(r-i))

pi =j≠i vj(rj,x(r-i)) -j≠i vj(rj,x) In Clarke mechanisms, agents’ utility are

always non-negative

Clarke mechanism for the Vickrey auction

The winner is paid pi = j≠i vj(rj,x(r-i)) -j≠i vj(rj,x) =

= j≠i vj(rj,x(r-i)) second lowest offer

Let us convince ourself it is strategy-proof by case analysis. For a player i, let T=minj≠i {rj}:

ti<T: then, if ri<ti, he still wins, but he keeps on to be paid T, while if ri>ti, he may still win (again being paid T), but he may also lose (if ri>T), by getting a null utility;

ti>T: then, if ri>ti, he keeps on not to win, while if ri<ti, he may win, but he will be paid T<ti, by getting a negative utility.

Remark: the difference between the second lowest offer and the lowest offer is unbounded (frugality issue)

VCG (Clarke)-Mechanisms: Pros and Cons

The goal, i.e., the optimization of the social-choice function, is achieved with certainty

The payments may be sub-optimal

VCG-mechanisms: algorithmic issues

System has to calculate N+1 functions: once with all agents (for g(r)) and once for every agent (for the associated payments)

What is the time complexity of the mechanism?

What happens if the problem is non-utilitarian?

What happens if computing the optimal solution is NP-hard?

VCG-mechanisms and graph problems

Following the Internet model, we assume that each agent owns a single edge of a graph G=(V,E), and establishes the cost for using it

The agent’s type is the true weight of the edge

Classic optimization problems on G become mechanism design optimization problems!

Many basic network design problems have been faced: minimum spanning tree, shortest paths, minimum Steiner tree, and many others

Let’s see what happens for shortest paths…

VCG-mechanism for the shortest path problem

Input: a selfish-edge undirected graph G (2-edge-connected), a source node s and a destination node t;

SCF: a true shortest path in G between s and t. VCG-mechanism: The problem is indeed

utilitarian! g(r): given G and the reported edge weights r=(r1,

…,rm), compute PG(s,t) pe: For any edge e PG(s,t), set pe=dG-e(s,t)-dG(s,t)+re

For any e PG(s,t), we have to compute PG-e(s,t), namely the replacement shortest path in G-e =(V,E\{e})) between s and t

Replacement shortest path

s

t

e

2

2

34

5 6

510

5

12

PG-e(s,t)PG(s,t)

A very efficient solution

Brute force: ePG(s,t), apply Dijkstra to find PG-

e(s,t) time complexity: O(mn + n2 logn) time In late 80’s, Malik et al. solved in O(m+n log n)

time the following vitality problem: given a shortest path PG(s,t), which is the most vital edge of it, i.e., the edge whose removal causes the longest replacement path between s and t?

Their approach consisted of computing all the replacement paths between s and t……but this is exactly what we are looking for in our VCG-mechanism!

Beyond VCG-mechanisms One-parameter mechanisms (Archer & Tardos,

SODA’01): provide strategy-proof mechanisms whenever the type of an agent is just a single parameter, as soon as; 1. the underlying algorithm is monotone (i.e., it keeps

on to (not) use an agent as soon as he decreases (increases) its reported type);

2. agents are paid w.r.t. their threshold value. One-parameter mechanisms are used in some classic

non-utilitarian problems (e.g., the Shortest-Paths Tree problem)

They are also used for several NP-hard problems, for which the fact that the underlying algorithm is sub-optimal prevents the use of a VCG-mechanism (e.g., the Steiner Tree problem)

Some (of our) results

inapprox2

Canonical Selfish-edge Selfish-node

SP O(m+n log n) O(m+n log n) O(m+n log n)

SPT O(m+n log n) O(m+n log n)

MST O(m (m,n)) O(m (m,n)) O(m (n,n))

MDST O(mn (m,n)) O(mn2 (m,n)) ?

MRST O(mn (m,n)) O(mn3/2 + n3)

ST 1.55-apx (2-2/k)-apx ?

inapprox133

135

Conclusions

AGT is a rapidly evolving discipline Many big questions are left open… …so, why not to try? (OR methods are

sought!) Suggested readings:

Algorithmic Game Theory, Edited by Noam Nisan, Tim Roughgarden, Eva Tardos, and Vijay V. Vazirani, Cambridge University Press.

Algorithmic Mechanism Design for Network Optimization Problems, Luciano Gualà, PhD Thesis, Università degli Studi dell’Aquila, 2007.

Web pages by Éva Tardos, Christos Papadimitriou, Tim Roughgarden, and then follow the links therein

Game over!