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

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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

- 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

- 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?

- 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

- 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!

- What are the computational aspects of a game?
- What does it mean to design an algorithm for a strategic distributed system?

Algorihmic

Game Theory

Theory of Algorithms

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

- 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)

- 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

Strategy

Set

Payoffs

Strategy Set

- 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:
- 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: 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

- 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

- 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)

- (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????

- 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!

- 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)

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!

- 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”

- 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!)

- 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)

- 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)

- 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!

- 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)

s

t

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

- 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.

- 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)

Assume i>>1

xi

1-

1

s

t

1

0

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

- 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?

- 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.

- 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.

- 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 )

- 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

Private “types”

Reported types

Mechanism

t1

r1

p1

Agent 1

Output

tn

rn

pn

Agent n

Payments

Each agent reports strategicallyto maximize its well-being…

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

- Algorithms Mechanisms
- Centralized Decentralized (Non-cooperative)
- Network design problems

- 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

- N agents, and each agent has some private information called thetype, tiTi, and performs a strategic action
- We restrict ourself to direct revelation mechanisms, in which the action is reporting a typeriTi(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)

- Example: Auction Game

- 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

- Example: Auction Game

- 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

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

- 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

- A VCG-mechanism is (the only) strategy-proof mechanism for utilitarianproblems:
- Algorithm:
g(r)= arg maxxFi 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

- Algorithm:
- What about non-utilitarian problems? We will see…

- 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.
■

- Payment given to agent i

- 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

- 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)

- The goal, i.e., the optimization of the social-choice function, is achieved with certainty
- The payments may be sub-optimal

- 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?

- 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…

- 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

s

PG(s,t)

3

2

PG-e(s,t)

4

5

6

5

e

2

10

12

5

t

- 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!

- 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

Game over!