Graphical Games: Equilibrium and Extensions 2.3 Graphical Games

Computational Economics （Algorithmic Game Theory）

Xiang-Yang Li, Lan Zhang School of Computer Science and Technology, USTC Professor, CS Department 2021 Graphical Games: Equilibrium and Extensions 2.3 Graphical Games

» 2.3.1 Graphical Games • compact representation » 2.3.2 Computing Nash Equilibria • TreeProp • NashProp : a distributed, message-passing algorithm » 2.3.3 Correlated Equilibria • definition and motivation • representation • computation

3 2.3 Graphical Games

4 Multiplayer Games » Large population games with limit players’ interaction. » Game Theory: • provide sound, rigorous mathematical formulation. • limited attention to problem representation: commonly “flat”, large-size representation that don’t exploit “structure”. • graph-based representation introduced to model interaction. » Multiplayer games Algorithms for computing equilibria in large population games with structured interactions Multiplayer Game Example » A multiplayer game consists of 푛 players, each with a finite set of pure strategies or actions available to them, along with a specification of the payoffs to each player. » Example 1: Pollution game • This game is the extension of Prisoner’s Dilemma to the case of many players. Different » Example 2: Tragedy of the commons. geographic locations Multiplayer Game Normal Form » A multiplayer game consists of n players, each with a finite set of pure strategies or actions available to them, along with a specification of the payoffs to each player.

» 푎푖 ∈ {0, 1} : the action chosen by player 푖, and the joint action 푎Ԧ ∈ {0, 1}푛

» 푀푖 ∈ [0, 1] : the payoff to player 푖, indexed by the joint action 푎Ԧ. » The actions 0 and 1 are the pure strategies of each player: • a mixed strategy for player 푖 is given by the probability 푝푖 ∈ [0, 1] that the player will play 0 » For any joint mixed strategy, given by a produce distribution 푝Ԧ, we define the expected payoff to player 푖 as : 푀푖 푝Ԧ = 퐸푎~푝Ԧ[푀푖(푎Ԧ)]

7 Issues with Normal Form

» The normal form representation of Multiplayer Game with large player population is not good enough. » Complexity: the number of parameters that must be specified grows exponentially with the size of the population. • Assuming 푛 players and 2 actions, as we have here, leads to the need for: 푛 matrices 푀푖(one for each player) each of size 2푛 » Conceptual: the normal form may fail to capture structure that is present in the strategic interaction: • Strong Influence • Weakly Influence

8 Structure in Multiplayer Game

» It is assumed that the payoff 푀푖 for player 푖 is a function of all the components 푎푗, (푗 = 1, … , 푛) in the joint action vector 푎Ԧ » However, the payoff for player 푖 may be dependent only on the actions of a subset of player 푁(푖) • Conditional independence payoff assumption » Perhaps: the payoff of each player are determined by the actions of only a small subpopulation. • Strong influence • The environmental pollution of a country is more affected by neighboring countries.

9 Graphical Model

» Graphical Model : models of structed probabilistic interaction. • 푛 points represent 푛 players, for which naïve representation is size : 2푛 • representation size mostly a function of the “degree of local interaction” among points or players. » Graph Model allows: • easy interpretation • compact representation • efficient computation in some cases Graphical Game & Graph Model

» Graphical Games adopt a simple Graph Model. » A graphical game is described by an undirected graph 퐺: • players ~ vertices » A player or vertex 푖 has payoffs that are entirely specified by the actions of 푖 and those of its neighbor set in 퐺 • Strong Influence along neighbors » Now the payoff to player 푖 is a function only of the actions of 푖 and its neighbors: • Rather than the actions of the entire population • Reduce Complexity.

11 Graphical Games

» Graphical games can capture and exploit locality or sparsity of direct influences.

» Graphical Games: • borrow representational ideas from graphical models. • intuitive graph interpretation: a player’s payoff is only a function of its neighborhood. • example : geography, networks • analogy to probabilistic graphical model : special structure. Graphical Games Formal Definition

» A graphical game is a pair (퐺, ℳ), where 퐺 is an undirected graph over the vertices {1, … , 푛}, and ℳ is a set of 푛 local game matrices. For any joint action 푎Ԧ, the 푖 local game matrix 푀푖 ∈ ℳ specifies the payoff 푀푖(푎 ) for player 푖, which depends only on the actions by the players in 푁(푖). Graphical Games Definition

» 퐺 : undirected graph representing the local interaction » 푀 : a set of 푛 local game matrices

» Player 푖’s payoff 푀푖 푎Ԧ is only a function of its neighborhood 푁(푖): implies conditional independence payoff assumption. ′ ′ » Local payoff matrix 푀푖 : 푀푖 푎Ԧ = 푀푖 푎Ԧ 푁 푖 » Graphical game complexity:

max degree of local interaction 푘 = 푚푎푥푖 푁 푖 ≪ 푛 representation size 푂(푛2푘) (exponential in max degree) Equilibrium Definition

» Nash equilibrium(NE): A Nash equilibrium for the game is a mixed strategy 푝Ԧ such that for any player 푖, and for any value ′ 푝푖 ∈ [0, 1] ′ 푀푖 푝Ԧ ≥ 푀푖(푝Ԧ[푖: 푝푖 ])

We say that 푝푖 is a best response to the rest of 푝Ԧ

» 휖-Nash equilibrium: An 휖-Nash equilibrium is a mixed strategy ′ 푝Ԧ such for any player 푖, and for any any value 푝푖 ∈ [0, 1] ′ 푀푖 푝Ԧ + 휖 ≥ 푀푖(푝Ԧ[푖: 푝푖 ])

We say that 푝푖 is an ε-best response to the rest of 푝Ԧ

15 2.3 Graphical Games

» 2.3.1 Graphical Games • compact representation » 2.3.2 Computing Nash Equilibria • TreeProp • NashProp » 2.3.3 Correlated Equilibria • definition and motivation • representation • computation

16 TreeNash Notation and Concept » TreeNash is a tree algorithm, which runs in polynomial time and provably computes approximations of all equilibria. » Capital letters: 푈, 푉 푎푛푑 푊 to denote player/vertex to distinguish them from their chosen actions/values(lower letter 푢, 푣, 푤).

» Local game matrix: 푀푉 to denote the local game matrix for the player/vertex 푉. TreeNash Notation and Concept » If G is a tree, we choose an arbitrary vertex as the root (which we visualize as being at the bottom, with the leaves Leaf/Parent at the top). » Upstream: any vertex on a path from 푉 to any leaf will be called upstream from 푉: downstream upstream • 푈푃퐺 푉 to denote the set of all vertices in 퐺 that are upstream from 푉

» Downstream: any vertex on the path Root/Child from a vertex 푉 to the root will be called downstream from 푉. TreeProp NE

» Suppose that 푉 is the child of 푈 in 퐺. » So, we let 퐺푈 denote the subgraph induced by the vertices in 푈푃퐺(푈)  the subtree of 퐺 rooted at 푈 » Let: 푣 ∈ [0, 1] is a mixed strategy for 푈 player/vertex 푉: ℳ푉=푣 will be the subset of payoff matrices in ℳ corresponding to the vertices in 푈푃퐺(푈) TreeProp NE

» An NE for the graphical game 푈 푈 (퐺 , ℳ푉=푣) is a conditional equilibrium “upstream” from 푈 – that is an equilibrium for 퐺푈 given that 푉 plays 푣.

» Here we are simply exploiting the fact that since 퐺 is a tree, fixing a mixed strategy 푣 for the play of 푉 isolates 퐺푈 from the rest of 퐺. TreeProp Downstream: 푇(푣, 푢) » Now suppose that vertex 푉 has 푘 parents 푈1, 푈2, … , 푈푘 and the single child 푊.

» The data structures sent from each 푈푖 to 푉, on the downstream pass of TreeNash , and in turn from 푉 to 푊 : • Each parent 푈 will send to 푉 a 푖 푇(푣, 푢푖) binary-valued “table” 푇(푣, 푢푖) TreeProp Downstream: 푇(푣, 푢) » The table is indexed by the continuum of possible values for the mixed strategies 푣 ∈ [0, 1] of 푉 and 푢푖 ∈ [0, 1] of 푈1, 푈2, … , 푈푘

» The semantics of this table: for any pair (푣, 푢 ), 푇(푣, 푢 ) will be 1 if and 푖 푖 푇(푣, 푢푖) only if there exists an NE for 푈 푈푖 푖 (퐺 , ℳ푉=푣) in which 푈1 = 푢푖 TreeProp Downstream: 푇(푤, 푣) » The data structures sent from 푉 to 푊, on the downstream pass of TreeNash » 푇(푤, 푣) = 1 ⟺ there exists a vector of mixed strategies 푢 = (푢1, … , 푢푘) (called a witness) for the parents 푈 = (푈1, … , 푈푘) of 푉 such that:

 푇 푣, 푢푖 = 1 for all 1 ≤ 푖 ≤ 푘  푉 = 푣 is a best response to 푈 = 푢, 푊 = 푤. 푇(푤, 푣) TreeProp Downstream » There may be more than one witness for 푇 (푤, 푣) = 1: • 푉 will also keep a list of the witnesses 푢 for each pair (푤, 푣) for which 푇 (푤, 푣) = 1 • These witness lists will be used on the upstream pass. » The downstream pass of the algorithm terminates at the root 푊. TreeProp Upstream » Begin: Root node W choose any witness (푉1, … , 푉푘) to 푇(푤) = 1, and then passing both w and 푣푖 to each parent 푉푖. • The interpretation is that 푊 will play 푤, and is “instructing” 푉푖 to play 푣푖 • if a vertex 푉 receives a value 푣

to play from its downstream 푇(푤, 푣) neighbor 푊, and the value 푤 that 푊 will play, then it must be that 푇 (푤, 푣) = 1. TreeProp Upstream » 푉 chooses a witness 푢 to 푇 (푤, 푣) = 1, and passes each parent 푈푖 their value 푢푖 as well as 푣

» Note that the semantics of 푇 (푤, 푣) =

1 ensure that 푉 = 푣 is a best response 푇(푣, 푢푖) to 푈 = 푢, 푊 = 푤. TreeProp Algorithm Flow » Downstream Pass:

• each node 푉 receives 푇(푣, 푢푖) from each 푈푖; • node 푉 computes 푇(푤, 푣) and witness lists for each 푇(푤, 푣) = 1 » Upstream Pass: • each node 푉 receives value (푤, 푣) from node W, s.t. 푇(푤, 푣) = 1 • node 푉 pick witness 푢 = (푢1, 푢2, … , 푢푘) for 푇(푤, 푣), pass (푣, 푢푖) to node 푈푖. TreeProp Algorithm Flow TreeProp Algorithm Flow TreeProp » Dynamic programming algorithm » Table-Passing phase • T(w, v) represent there exists a NE “upstream” in which 푉 plays 푣 and 푊 is “clamped” to 푤. » Assignment-Passing phase • assign NE mixed strategy to root. - downstream • recursively find assignment for immediate “upstream” neighbors consistent with tables. TreeProp » Representation results • for ϵ − NE need τ − size grid for tables polynomial in model size and 1/ϵ

» Lemma: Let (퐺, ℳ) be any graphical game in which 퐺 is a tree. Algorithm TreeNash computes a Nash equilibrium for (퐺, ℳ). Furthermore, the tables and witness lists computed by the algorithm represent all Nash equilibria of (퐺, ℳ). TreeProp: A Slight Issue » The actions (푢, 푣, 푤 … ) are continuous variables → 푇(푤, 푣) be represented compactly? Approximate TreeNash » Discretization of the action space

» Player I can now only play action 푞푖 ∈ {1, 휏, 2휏, … , 1} » Algorithm takes an extra input parameter 휖 » At each node the 휖-best response is computed(휏 = 푂(휖/푑))

» Lemma: Approximate TreeNash computes a 휖-NE for the game (퐺, 푀) in time polynomial in the representation of (퐺, 푀) Exact TreeNash » However its complexity is exponential in the number of vertices of 퐺.

» Computing an exact equilibrium in time polynomial in the size of the tree remains an open issue. NashProp for Arbitrary Graph

» A distributed, message-passing algorithm: natural extension of TreeProp to arbitrary graphs. » each 푉 does this computation once for each of its neighbors: • each time “pretending” that this neighbor plays the role of the downstream neighbor in TreeNash; • the remaining neighbors play the roles of the upstream 푈푖. Table-Passing

» Table-Passing phase: • each 푉 does this computation once for each of its neighbors • the remaining neighbors play the roles of the upstream 푈푖 • Initialization: 푇 0 푤, 푣 = 1 for all (푤, 푣) • Induction: 푇 푟 + 1 푤, 푣 = 1, 푖푓푓 ∃ 푢: » 푇 푟 푣, 푢푖 = 1 푓표푟 푎푙푙 푖 » 푉 = 푣 a best response for 푊 = 푤, 푈 = 푢 Convergence of Table-Passing

» Table-Passing phase always converges. » ALL NE preserved. » For discretization scheme: • tables converge quickly(number of rounds polynomial in model size). • each round takes polynomial in model size(for fixed grid size). Assignment-Passing

» NEs preserved but search still needed » More 0’s in tables can lead to significantly reduced search space. » Many heuristics possible • Backtracking local search » Discretization scheme: Computation time per round polynomial in model size (for fixed grid size). • Discretization scheme leads to constraint satisfaction problem (CSP) formulations. • NashProp is a particular instantiation of arc-consistency followed by backtracking local search in a particular CSP. NashProp Summary

» Converging first phase with • table size for 휖 − 푁퐸 polynomial in the size of the model. • running time also polynomial (for fixed k). » Second phase is backtracking local search. » For both phases: each round polynomial in the size of the model (for fixed k) Related Work

» Exact NE Computation  All NE in trees: exponential in representation size [Kearns, Littman and Singh, 2001]; Single NE in trees: polynomial for 2-action [Littman, Kearns and Singh, 2002], m-action open!; Single NE in loopy graphs: continuation-method heuristic [Blum, Shelton and Koller, 2003] » Other approximation heuristics  CSP formulation: Cluster [Kearns, Littman and Singh, 2001] and junction- tree [Vickrey and Koller, 2002]; Gradient ascent and “hybrid” approaches [Vickrey and Koller, 2002] » Some results on computing NE for torus-like GG [Daskalakis and Papadimitriou, 2004] Summary

» NashProp for computing Nash equilibria in graphical games : • Distributed, message-passing: Avoids centralized computation • Runs directly on game graph: Avoids operating on “hyper- graphs” • Generalizes approximation algorithm for trees • Strong theoretical guarantees for table-passing phase • Assignment passing phase: Simple implementations experimentally sufficient and effective in arbitrary graphs • Promising, effective heuristic in loopy graphs (as loopy belief propagation) 2.6 Graphical Games

» 2.6.1 Graphical Games • compact representation » 2.6.2 Computing Nash Equilibria • TreeProp • NashProp : a distributed, message-passing algorithm » 2.6.3 Correlated Equilibria • definition and motivation • representation • computation

43 Example: Traffic Light

» The game we consider is when two players drive up to the same intersection at the same time. If both attempt to cross, the result is a fatal traffic accident. The game can be modeled by a payoff matrix where crossing successfully has a payoff of 1, not crossing pays 0, while an accident costs −100. Example: Traffic Light

» This game has three Nash equilibria: • two correspond to letting only one car cross. • the third is a mixed equilibrium where both players cross 1 with an extremely small probability 휖 = , and with 휖2 101 probability they crash. » In a correlated equilibrium a coordinator can choose strategies for both players • For example, the coordinator can randomly let one of the two players cross with any probability. • The player who is told to stop has 0 payoff, but he knows that attempting to cross will cause a traffic accident. Correlated Equilibria

» Same setting as before: Games with greedy players. » Mathematical formulation: Normal-form games : • A set of players {1, … , 푛}, each with a set of actions or pure strategies 퐴 = {1, … , 푚}, joint-action 푛 (푎1, … , 푎푛) ∈ 퐴

• Payoff matrix 푀푖: player i’s payoff 푀푖(푎1, … , 푎푛) » Correlated equilibrium (CE): A joint probability distribution 푃(푎1, … , 푎푛) such that: • Every player individually receives “suggestion” from P. • Knowing P , players are happy with “suggestion” » NE as a special case: P - a product distribution; always exists! Correlated Equilibria

» Definition: A correlated equilibrium (CE) for a two-action normal form game is distribution 푃(푎Ԧ) over actions satisfying:

∀푖 ∈ 1, … , 푛 , ∀푏 ∈ 0,1 : 퐸푎~푃 [푀푖(푎Ԧ)] ≥ 퐸푎~푃 [푀푖(푎Ԧ[푖: −푏])] 푎푖=푏 푎푖=푏 CE in Graphical Games

» Representation

• P(a1, … , an) exponential in number of players 푛 • Does succinct graphical game representation? ⇒ succinct CE representation? (Intuition: interaction due to game should govern correlations) • Yes (under a reasonable equivalence class) » Computation • Normal-form games: CE computable via LP (with variables P(a1, … , an) ). • Does succinct graphical game computation? ⇒ efficient CE algorithm? • Yes (for some interesting subclass of graphical games) Representation - succinctly

» First, we argue that it is not necessary to model all the correlations that might arise in a CE, but only those required to represent all of the possible (expected payoff) outcomes for the players. » Second, we need show that the remaining correlations can be represented by a Markov network. • the interactions between vertices in the graphical game are entirely strategic and given by local payoff matrices • the interactions in the associated Markov network are entirely probabilistic and given by local potential functions CE Equivalence

» Representation issue: In general, just considering arbitrary CE does not help representationally • Want to preserve succinctness of GG in CE representation as well

» Keypoint: there is a natural subclass of the set of all CE of a graphical game, based on expected payoff equivalence, whose representation size is linearly related to the representation size of the graphical game. EPE & LNE Definition

» EPE: two distribution 푃 and 푄 over joint actions 푎Ԧ are expected payoff equivalent, denote 푃 ≡퐸푃 푄, if 푃 and 푄 yield the same expected payoff vector: for each 푖, 퐸푎~푃 푀푖 푎Ԧ = 퐸푎~푄 푀푖 푎Ԧ • Expected payoff equivalence of two distributions is dependent upon the reward matrices of a graphical game » LNE: For a graph 퐺, two distribution 푃 and 푄 over joint actions 푎Ԧ are local neighborhood equivalent with respect to 푖 퐺, denote 푃 ≡퐿푁 푄, if for all players 푖, and for all setting 푎Ԧ of 푁(푖), 푃 푎Ԧ푖 = 푄(푎Ԧ푖) • the marginal distributions over all local neighborhoods is dependent up on the context of graph 퐺. EPE & LNE Equation Lemma

» Lemma: For all graphs 퐺, for all joint distributions 푃 and 푄 on actions, and for all graphical games with graph 퐺, if 푃 ≡퐿푁 푄 then 푃 ≡퐸푃 푄. Furthermore, for any graph 퐺 and joint distributions 푃 and 푄, there exist payoff matrices ℳ such that for the graphical (퐺, ℳ), if 푃 ≢퐿푁 푄 then 푃 ≢퐸푃 푄. LNE ⇒ EPE

Thus local neighborhood equivalence implies payoff equivalence, but the converse is not true in general. We now establish that local neighborhood equivalence also preserves CE. Efficient CE Representation Lemma

» Lemma: For any graphical game (퐺, ℳ), if 푃 is a CE for (퐺, ℳ) and 푃 ≡퐿푁 푄 then 푄 is a CE for (퐺, ℳ).

» Now we can concisely represent, in a single model, all CE up to local neighborhood (and therefore payoff) equivalence. Correlated Equilibria & Markov Nets

» Graphical games provide a concise language for expressing local interaction in game theory » Markov networks exploit undirected graphs for expressing local interaction in probability distributions » Markov networks are a natural and powerful language for expressing the CE of a graphical game: • there is a close relationship between the graph of the game and its associated Markov network graph Local Markov Nets Definition

» A local Markov network is a pair M ≡ (G, Ψ), where • 퐺 is an undirected graph on vertices {1, … , 푛} • Ψ is a set of potential functions, one for each local neighborhood 푁(푖), mapping binary assignments of values of 푁(푖) to the range [0, ∞): 푖 Ψ ≡ {휓푖: 푖 = 1, … , 푛; 휓푖: {푎Ԧ } → [0, ∞)}, where {푎Ԧ푖}is the set of all 2|푁(푖)| settings to 푁(푖) Local Markov Networks

» Compact representation of joint probability distributions » Graph G = (V,E) • vertices V correspond to random variables • potential function for each neighborhood of G 푖 ∀푖, 휓푖: 푎Ԧ → [0,+∝) » Local Markov network: M ≡ (G, Ψ) » A local Markov network 푀 defines a probability distribution 푃 as 푛 1 푃 푎Ԧ = ෑ 휓 (푎Ԧ푖) 푍 푖 푖=1 » Again, representation size exponential in size of largest neighborhood CE Representation Lemma

» Lemma: for all graphs 퐺, and for all joint distribution 푃 over joint actions, there exist a distribution 푄 that is representable as a local Markov network with graph 퐺 such that Q ≡퐿푁 푃 with respect to 퐺. » Lemma: for all graphical games (퐺, ℳ), and for any correlated equilibrium 푃 of (퐺, ℳ), there exist a distribution 푄 such that: • 푄 is also correlated equilibrium for (퐺, ℳ)

• 푄 gives all players the same expected payoffs as P: Q ≡퐸푃 푃 • 푄 can be represented as a local Markov network with graph 퐺. CE Representation Lemma

» Efficient CE Representation Lemma : For every CE 푃 for a graphical game with graph 퐺, there exists a CE 푄 for the game with the properties • 푄 is a local Markov network with graph 퐺 • Q is expected payoff equivalent to P » Proof idea/sketch: Maximum entropy (ME) distribution consistent with local neighborhood distributions of P satisfies those conditions » Implication: Qualitative probabilistic properties of CEs in GGs Computation: Normal-form Games

» CE conditions correspond to linear inequalities in the joint distribution values 푃(푎1, … , 푎푛)

» Distribution constraints also linear in 푃(푎1, … , 푎푛) » Known result: We can compute a single exact CE for a game in normal-form in time polynomial in the representation size of the game (푂(2푛)) by using linear programming (LP) » Can we preserve succinctness of GG representation in CE computation? Constraints

푖 » Variables: {푃푖(푎Ԧ )} (local neighborhood marginals) » Global CE constraints (can be a large set) 푖 • Best response: linear in {푃푖(푎Ԧ )} 푖 • Global consistency: ensure {푃푖(푎Ԧ )} correspond to some proper global joint probability distribution P (a1, . . . , an) » Local CE constraints (polynomial number of linear constraints) • Best response • Local Marginal Distribution • Intersection Consistency » In general, local consistency does not imply global consistency, BUT ... Constraints

» Local consistency sufficient for global consistency if the game graph is a tree (for example) » Efficient Tree Algorithm : • There exists an algorithm (based on LP) that finds a CE 푄 for tree graphical games with graph 퐺 in time polynomial in the representation size of the game » Can be extended to bounded tree-width graphical games » 푄 is also a local Markov network with graph 퐺. » Can sample uniformly from set of CE in polynomial-time for bounded tree-width GG Linear Programming

» Variables: for every player I and assignment 푎Ԧ푖, there is a variable 푃(푎Ԧ푖) » LP Constraints: • CE Constraints: for all players 푖 and actions 푎, 푎′ 푖 푖 푖 푖 ′ ෍ 푃(푎Ԧ )푀푖(푎Ԧ ) ≥ ෍ 푃 푎Ԧ 푀푖[푎Ԧ [푖: 푎 ]] 푖 푖 푖 푖 푎 :푎푖=푎 푎 :푎푖=푎 • Neighborhood Marginal Constraints: for all players 푖: ∀푎Ԧ푖: 푃 푎Ԧ푖 ≥ 0, ෍ 푃 푎Ԧ푖 = 1 푎푖 • Intersection Consistency Constraints: for all players 푖 and 푗, and for any assignment 푦Ԧ푖푗 to the intersection 푁(푖) ∩ 푁(푗) : 푖푗 푖 푗 푖푗 푃 푎Ԧ ≡ ෍ 푃 푎Ԧ = ෍ 푃푗 푎Ԧ ≡ 푃푗(푎Ԧ ) 푎푖:푎푖푗=푦푖푗 푎푗:푎푖푗=푦푖푗 Algorithm Result Lemma

» Lemma: for all tree graphical games (퐺, ℳ), any solution to the linear constraints given above is a correlated equilibrium for (퐺, ℳ) CE in GG: Summary

» CE Representation Theorem • Every CE for a graphical game can be characterized by another achieving the same expected payoffs vector and which can be represented in size polynomial in the representation size of the game

» Efficient Tree Algorithm (Generalizes normal-form games algorithm) • For all graphical games with graphs in a certain class, containing trees and full-graphs as “canonical” examples, we can compute a CE for the game in time polynomial in the representation size of the game Open Problems/Areas

» Efficient Algorithms for Exact Nash Computation in Trees. » Strategy-Proof Algorithms for Distributed Nash Computation. » Cooperative, Behavioral, and Other Equilibrium Notions. THANKS!

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