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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?

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algorithmic game theory because a game is nice when it s not too long

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

  • 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

Nash equilibria

two research traditions
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
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
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
Fundamental Questions
  • What are the computational aspects of a game?
  • What does it mean to design an algorithm for a strategic distributed system?


Game Theory

Theory of Algorithms

Game Theory



game 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
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 siSi(nobody can observe others’ move)
  • 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
prisoner i s decision
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 ii s decision
Prisoner II’s decision
  • Prisoner II’s decision:
    • If I chooses Don’t Implicatethen it is best to Implicate
    • If I chooses Implicatethen it is best to Implicate
    • It is best to Implicate for II, regardless of what I does: Dominant Strategy
  • 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
dominant strategy equilibrium
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
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
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
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????

a conflictual game head or tail
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!

three big computational issues
Three big computational issues
  • Finding a NE, once it does exist
  • 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)
  • 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
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
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
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 P is NP-hard if one can reduce any NP-complete problem P’ to it:
    • “yes”-instance of P’ → “yes”-instance of P
    • “no”-instance of P’→ “no”-instance of P
  • 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
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=PNP…

…but several PPAD-complete problems are resisting for decades to poly-time attacks (e.g., finding Brouwer fixed points)

2 nash is ppad complete
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
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
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

A case study: selfish routing on Internet

  • 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???

Example: Pigou’s game (network congestion game)

Latency depends on the congestion (x is the fraction of flow using the edge)



Latency is fixed

  • 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
flows and ne
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
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
A bad example for non-linear latencies

Assume i>>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
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
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
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 gameG 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
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!


Algorithmic Mechanism Design

(or, the art of convincing a capitalist to behave like a socialist )

mechanism design the goal
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
Mechanism Design: a picture

Private “types”

Reported types





Agent 1





Agent n


Each agent reports strategicallyto maximize its well-being…

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

overview of the results
Overview of the results
  • Algorithms  Mechanisms
  • Centralized  Decentralized (Non-cooperative)
    • Network design problems
mechanism design
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
Mechanism Design Problem: Type of an Agent
  • N agents, and each agent has some private information called thetype, tiTi, and performs a strategic action
  • We restrict ourself to direct revelation mechanisms, in which the action is reporting a typeriTi(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
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
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
The Mechanism

A mechanismis a pairM=<x=g(r), p(x)> specifying:

  • An algorithm g(r) which computes the outcome x as a function of the reported typesr
  • A paymentscheme p as a function of the output x
strategy proof mechanisms
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
Vickrey-Clarke-Groves (VCG) Mechanisms
  • A VCG-mechanism is (the only) strategy-proof mechanism for utilitarianproblems:
    • Algorithm:

g(r)= arg maxxFi vi(ri,x)

    • Payment function:

pi (x)= hi(r-i) -j≠i vj(rj,x)

wherehi(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
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
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
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
VCG (Clarke)-Mechanisms: ProsandCons
  • 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
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
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
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
Replacement shortest path
















a very efficient solution
A very efficient solution
  • Brute force:  ePG(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
Beyond VCG-mechanisms
  • One-parametermechanisms (Archer & Tardos, SODA’01): provide strategy-proof mechanisms whenever the type of an agent is just a single parameter, as soon as;
    • the underlying algorithm is monotone(i.e., it keeps on to (not) use an agent as soon as he decreases (increases) its reported type);
    • 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)
  • 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