computational economics algorithmic game theory

Computational Economics（Algorithmic Game Theory）

Xiang-Yang Li, Lan ZhangSchool of Computer Science and Technology, USTC

Professor, CS Department2021

Graphical Games:Equilibrium and Extensions

» 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

2.3 Graphical Games

3




• TreeProp




• representation

• computation

2.3 Graphical Games

4

» 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

Multiplayer Games

Algorithms for computing equilibria in large population games with structured interactions

» 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.

» Example 1: Pollution game

• This game is the extension of Prisoner’s Dilemma to the case of many players.

» Example 2: Tragedy of the commons.

Multiplayer Game Example

Different geographic locations

» 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.

» 𝑎𝑖 ∈ {0, 1} : the action chosen by player 𝑖, and the joint action Ԧ𝑎 ∈ {0, 1}𝑛

» 𝑀𝑖 ∈ [0, 1] : the payoff to player 𝑖, indexed by the joint action Ԧ𝑎.

» The actions 0 and 1 are the pure strategies of each player:

• a mixed strategy for player 𝑖 is given by the probability 𝑝𝑖 ∈ [0, 1] that the player will play 0

» For any joint mixed strategy, given by a produce distribution Ԧ𝑝, we define the expected payoff to player 𝑖 as :

𝑀𝑖 Ԧ𝑝 = 𝐸𝑎~ Ԧ𝑝[𝑀𝑖( Ԧ𝑎)]

Multiplayer Game Normal Form

7

» 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 𝑛 players and 2 actions, as we have here, leads to the need for: 𝑛 matrices 𝑀𝑖(one for each player) each of size 2𝑛

» Conceptual: the normal form may fail to capture structurethat is present in the strategic interaction:

• Strong Influence

• Weakly Influence

Issues with Normal Form

8

» It is assumed that the payoff 𝑀𝑖 for player 𝑖 is a function of all the components 𝑎𝑗 , (𝑗 = 1,… , 𝑛) in the joint action

vector Ԧ𝑎

» However, the payoff for player 𝑖 may be dependent only on the actions of a subset of player 𝑁(𝑖)

• 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.

Structure in Multiplayer Game

9

» Graphical Model : models of structed probabilistic interaction.

• 𝑛 points represent 𝑛 players, for which naïve representation is size : 2𝑛

• representation size mostly a function of the “degree of local interaction” among points or players.

» Graph Model allows:

• easy interpretation


• efficient computation in some cases

Graphical Model

» Graphical Games adopt a simple Graph Model.

» A graphical game is described by an undirected graph 𝐺:

• players ~ vertices

» A player or vertex 𝑖 has payoffs that are entirely specified by the actions of 𝑖 and those of its neighbor set in 𝐺

• Strong Influence along neighbors

» Now the payoff to player 𝑖 is a function only of the actions of 𝑖 and its neighbors:

• Rather than the actions of the entire population

• Reduce Complexity.

Graphical Game & Graph Model

11

» 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

Graphical Games Formal Definition

» A graphical game is a pair (𝐺,ℳ), where 𝐺 is an undirected graph over the vertices {1,… , 𝑛}, and ℳ is a set of 𝑛 local game matrices. For any joint action Ԧ𝑎, the

local game matrix 𝑀𝑖 ∈ ℳ specifies the payoff 𝑀𝑖(𝑎𝑖) for

player 𝑖, which depends only on the actions by the players in 𝑁(𝑖).

Graphical Games Definition

» 𝐺 : undirected graph representing the local interaction

» 𝑀 : a set of 𝑛 local game matrices

» Player 𝑖’s payoff 𝑀𝑖 Ԧ𝑎 is only a function of its neighborhood 𝑁(𝑖): implies conditional independence payoff assumption.

» Local payoff matrix 𝑀𝑖′: 𝑀𝑖 Ԧ𝑎 = 𝑀𝑖

′ Ԧ𝑎 𝑁 𝑖

» Graphical game complexity:

max degree of local interaction 𝑘 = 𝑚𝑎𝑥𝑖 𝑁 𝑖 ≪ 𝑛representation size 𝑂(𝑛2𝑘) (exponential in max degree)

» Nash equilibrium(NE): A Nash equilibrium for the game is a mixed strategy Ԧ𝑝 such that for any player 𝑖, and for any value 𝑝𝑖′ ∈ [0, 1]

𝑀𝑖 Ԧ𝑝 ≥ 𝑀𝑖( Ԧ𝑝[𝑖: 𝑝𝑖′])

We say that 𝑝𝑖 is a best response to the rest of Ԧ𝑝

» 𝜖-Nash equilibrium: An 𝜖-Nash equilibrium is a mixed strategy Ԧ𝑝 such for any player 𝑖, and for any any value 𝑝𝑖

′ ∈ [0, 1]𝑀𝑖 Ԧ𝑝 + 𝜖 ≥ 𝑀𝑖( Ԧ𝑝[𝑖: 𝑝𝑖

′])

We say that 𝑝𝑖 is an ε-best response to the rest of Ԧ𝑝

Equilibrium Definition

15




• TreeProp

• NashProp



• representation

• computation

2.3 Graphical Games

16

» TreeNash is a tree algorithm, which runs in polynomial time and provably computes approximations of all equilibria.

» Capital letters: 𝑈, 𝑉 𝑎𝑛𝑑 𝑊 to denote player/vertex to distinguish them from their chosen actions/values(lower letter 𝑢, 𝑣, 𝑤).

» Local game matrix: 𝑀𝑉 to denote the local game matrix for the player/vertex 𝑉.

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 at the top).

» Upstream: any vertex on a path from 𝑉 to any leaf will be called upstreamfrom 𝑉:

• 𝑈𝑃𝐺 𝑉 to denote the set of all vertices in 𝐺 that are upstream from 𝑉

» Downstream: any vertex on the path from a vertex 𝑉 to the root will be called downstream from 𝑉.

TreeNash Notation and Concept

Leaf/Parent

Root/Child

upstreamdownstream

» Suppose that 𝑉 is the child of 𝑈 in 𝐺.

» So, we let 𝐺𝑈 denote the subgraph induced by the vertices in 𝑈𝑃𝐺(𝑈)

the subtree of 𝐺 rooted at 𝑈

» Let: 𝑣 ∈ [0, 1] is a mixed strategy for player/vertex 𝑉: ℳ𝑉=𝑣

𝑈 will be the subset of payoff matrices in ℳ corresponding to the vertices in 𝑈𝑃𝐺(𝑈)

TreeProp NE

» An NE for the graphical game (𝐺𝑈,ℳ𝑉=𝑣

𝑈 ) is a conditional equilibrium “upstream” from 𝑈 – that is an equilibrium for 𝐺𝑈 given that 𝑉 plays 𝑣.

» Here we are simply exploiting the fact that since 𝐺 is a tree, fixing a mixed strategy 𝑣 for the play of 𝑉 isolates 𝐺𝑈

from the rest of 𝐺.

TreeProp NE

» Now suppose that vertex 𝑉 has 𝑘parents 𝑈1, 𝑈2, … , 𝑈𝑘 and the single child 𝑊.

» The data structures sent from each 𝑈𝑖to 𝑉, on the downstream pass of TreeNash , and in turn from 𝑉 to 𝑊 :

• Each parent 𝑈𝑖 will send to 𝑉 a binary-valued “table” 𝑇(𝑣, 𝑢𝑖)

TreeProp Downstream: 𝑇(𝑣, 𝑢)

𝑇(𝑣, 𝑢𝑖)

» The table is indexed by the continuumof possible values for the mixed strategies 𝑣 ∈ [0, 1] of 𝑉 and 𝑢𝑖 ∈[0, 1] of 𝑈1, 𝑈2, … , 𝑈𝑘

» The semantics of this table: for any pair (𝑣, 𝑢𝑖), 𝑇(𝑣, 𝑢𝑖) will be 1 if and only if there exists an NE for

(𝐺𝑈𝑖 ,ℳ𝑉=𝑣𝑈𝑖 ) in which 𝑈1 = 𝑢𝑖

TreeProp Downstream: 𝑇(𝑣, 𝑢)


» The data structures sent from 𝑉 to 𝑊, on the downstream pass of TreeNash

» 𝑇(𝑤, 𝑣) = 1⟺ there exists a vector of mixed strategies 𝑢 =(𝑢1, … , 𝑢𝑘) (called a witness) for the

parents 𝑈 = (𝑈1, … , 𝑈𝑘) of 𝑉 such that:

𝑇 𝑣, 𝑢𝑖 = 1 for all 1 ≤ 𝑖 ≤ 𝑘

𝑉 = 𝑣 is a best response to 𝑈 =𝑢,𝑊 = 𝑤.

TreeProp Downstream: 𝑇(𝑤, 𝑣)

𝑇(𝑤, 𝑣)

» There may be more than one witness for 𝑇 (𝑤, 𝑣) = 1:

• 𝑉 will also keep a list of the witnesses 𝑢 for each pair (𝑤, 𝑣) for which 𝑇 (𝑤, 𝑣) = 1

• These witness lists will be used on the upstream pass.

» The downstream pass of the algorithm terminates at the root 𝑊.

TreeProp Downstream

» Begin: Root node W choose any witness (𝑉1, … , 𝑉𝑘) to 𝑇(𝑤) = 1, and then passing both w and 𝑣𝑖 to each parent 𝑉𝑖.

• The interpretation is that 𝑊 will play 𝑤, and is “instructing” 𝑉𝑖 to play 𝑣𝑖

• if a vertex 𝑉 receives a value 𝑣to play from its downstream neighbor 𝑊, and the value 𝑤that 𝑊 will play, then it must be that 𝑇 (𝑤, 𝑣) = 1.

TreeProp Upstream

𝑇(𝑤, 𝑣)

» 𝑉 chooses a witness 𝑢 to 𝑇 (𝑤, 𝑣) = 1, and passes each parent 𝑈𝑖 their value 𝑢𝑖 as well as 𝑣

» Note that the semantics of 𝑇 (𝑤, 𝑣) =1 ensure that 𝑉 = 𝑣 is a best response

to 𝑈 = 𝑢,𝑊 = 𝑤.

TreeProp Upstream


» Downstream Pass:

• each node 𝑉 receives 𝑇(𝑣, 𝑢𝑖)from each 𝑈𝑖;

• node 𝑉 computes 𝑇(𝑤, 𝑣) and witness lists for each 𝑇(𝑤, 𝑣) = 1

» Upstream Pass:

• each node 𝑉 receives value (𝑤, 𝑣) from node W, s.t. 𝑇(𝑤, 𝑣) =1

• node 𝑉 pick witness 𝑢 =(𝑢1, 𝑢2, … , 𝑢𝑘) for 𝑇(𝑤, 𝑣), pass (𝑣, 𝑢𝑖) to node 𝑈𝑖.

TreeProp Algorithm Flow

TreeProp Algorithm Flow

» Dynamic programming algorithm

» Table-Passing phase

• T(w, v) represent there exists a NE “upstream” in which 𝑉 plays 𝑣 and 𝑊 is “clamped” to 𝑤.

» 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 (𝐺,ℳ) be any graphical game in which 𝐺 is a tree. Algorithm TreeNash computes a Nash equilibrium for (𝐺,ℳ). Furthermore, the tables and witness lists computed by the algorithm represent all Nash equilibria of (𝐺,ℳ).

TreeProp

» The actions (𝑢, 𝑣, 𝑤 … ) are continuous variables → 𝑇(𝑤, 𝑣) be represented compactly?

TreeProp: A Slight Issue

» Discretization of the action space

» Player I can now only play action 𝑞𝑖 ∈ {1, 𝜏, 2𝜏, … , 1}

» Algorithm takes an extra input parameter 𝜖

» At each node the 𝜖-best response is computed(𝜏 =𝑂(𝜖/𝑑))

» Lemma: Approximate TreeNash computes a 𝜖-NE for the game (𝐺,𝑀) in time polynomial in the representation of (𝐺,𝑀)

Approximate TreeNash

» However its complexity is exponential in the number of vertices of 𝐺.

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

Exact TreeNash

» A distributed, message-passing algorithm: natural extension of TreePropto arbitrary graphs.

» each 𝑉 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 𝑈𝑖.

NashProp for Arbitrary Graph

» Table-Passing phase:

• each 𝑉 does this computation once for each of its neighbors

• the remaining neighbors play the roles of the upstream 𝑈𝑖

• Initialization: 𝑇 0 𝑤, 𝑣 = 1 for all (𝑤, 𝑣)

• Induction: 𝑇 𝑟 + 1 𝑤, 𝑣 =1, 𝑖𝑓𝑓 ∃ 𝑢:» 𝑇 𝑟 𝑣, 𝑢𝑖 = 1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖» 𝑉 = 𝑣 a best response for

𝑊 = 𝑤,𝑈 = 𝑢

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).

Convergence of Table-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.

Assignment-Passing

» Converging first phase with

• table size for 𝜖 − 𝑁𝐸 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)

NashProp Summary

» 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]

Related Work

» 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)

Summary




• TreeProp




• representation

• computation

2.6 Graphical Games

43

» 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

with an extremely small probability 𝜖 =1

101, and with 𝜖2

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.

Example: Traffic Light

» Same setting as before: Games with greedy players.

» Mathematical formulation: Normal-form games :

• A set of players {1,… , 𝑛}, each with a set of actions or pure strategies 𝐴 = {1,… ,𝑚}, joint-action (𝑎1, … , 𝑎𝑛) ∈ 𝐴

𝑛

• Payoff matrix 𝑀𝑖: player i’s payoff 𝑀𝑖(𝑎1, … , 𝑎𝑛)

» Correlated equilibrium (CE): A joint probability distribution 𝑃(𝑎1, … , 𝑎𝑛) 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

Correlated Equilibria

» Definition: A correlated equilibrium (CE) for a two-actionnormal form game is distribution 𝑃( Ԧ𝑎) over actions satisfying:

∀𝑖 ∈ 1, … , 𝑛 , ∀𝑏 ∈ 0,1 : 𝐸𝑎~𝑃𝑎𝑖=𝑏[𝑀𝑖( Ԧ𝑎)] ≥ 𝐸𝑎~𝑃𝑎𝑖=𝑏

[𝑀𝑖( Ԧ𝑎[𝑖: −𝑏])]

» Representation

• P(a1, … , an) exponential in number of players 𝑛

• 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)

CE in Graphical Games

» 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

Representation - succinctly

» 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.

CE Equivalence

EPE & LNE Definition

» EPE: two distribution 𝑃 and 𝑄 over joint actions Ԧ𝑎 are expected payoff equivalent, denote 𝑃 ≡𝐸𝑃 𝑄, if 𝑃 and 𝑄yield the same expected payoff vector: for each 𝑖, 𝐸𝑎~𝑃 𝑀𝑖 Ԧ𝑎 = 𝐸𝑎~𝑄 𝑀𝑖 Ԧ𝑎

• Expected payoff equivalence of two distributions is dependent upon the reward matrices of a graphical game

» LNE: For a graph 𝐺, two distribution 𝑃 and 𝑄 over joint actions Ԧ𝑎 are local neighborhood equivalent with respect to 𝐺, denote 𝑃 ≡𝐿𝑁 𝑄, if for all players 𝑖, and for all setting Ԧ𝑎𝑖

of 𝑁(𝑖), 𝑃 Ԧ𝑎𝑖 = 𝑄( Ԧ𝑎𝑖)

• the marginal distributions over all local neighborhoods is dependent up on the context of graph 𝐺.

EPE & LNE Equation Lemma

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.

» Lemma: For all graphs 𝐺, for all joint distributions 𝑃 and 𝑄on actions, and for all graphical games with graph 𝐺, if 𝑃 ≡𝐿𝑁 𝑄 then 𝑃 ≡𝐸𝑃 𝑄. Furthermore, for any graph 𝐺 and joint distributions 𝑃 and 𝑄, there exist payoff matrices ℳsuch that for the graphical (𝐺,ℳ), if 𝑃 ≢𝐿𝑁 𝑄 then 𝑃 ≢𝐸𝑃 𝑄.

LNE ⇒ EPE

Efficient CE Representation Lemma

» Lemma: For any graphical game (𝐺,ℳ), if 𝑃 is a CE for (𝐺,ℳ) and 𝑃 ≡𝐿𝑁 𝑄 then 𝑄 is a CE for (𝐺,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

• 𝐺 is an undirected graph on vertices {1, … , 𝑛}

• Ψ is a set of potential functions, one for each local neighborhood 𝑁(𝑖), mapping binary assignments of values of 𝑁(𝑖) to the range [0,∞):

Ψ ≡ {𝜓𝑖: 𝑖 = 1,… , 𝑛; 𝜓𝑖: { Ԧ𝑎𝑖} → [0,∞)},

where { Ԧ𝑎𝑖}is the set of all 2|𝑁(𝑖)| settings to 𝑁(𝑖)

» Compact representation of joint probability distributions

» Graph G = (V,E)

• vertices V correspond to random variables

• potential function for each neighborhood of G

∀𝑖, 𝜓𝑖: Ԧ𝑎𝑖 → [0,+∝)

» Local Markov network: M ≡ (G,Ψ)

» A local Markov network 𝑀 defines a probability distribution 𝑃 as

𝑃 Ԧ𝑎 =1

𝑍ෑ

𝑖=1

𝑛

𝜓𝑖( Ԧ𝑎𝑖)

» Again, representation size exponential in size of largest neighborhood

Local Markov Networks

CE Representation Lemma

» Lemma: for all graphs 𝐺, and for all joint distribution 𝑃 over joint actions, there exist a distribution 𝑄 that is representable as a local Markov network with graph 𝐺 such that Q ≡𝐿𝑁 𝑃with respect to 𝐺.

» Lemma: for all graphical games (𝐺,ℳ), and for any correlated equilibrium 𝑃 of (𝐺,ℳ), there exist a distribution 𝑄 such that:

• 𝑄 is also correlated equilibrium for (𝐺,ℳ)

• 𝑄 gives all players the same expected payoffs as P: Q ≡𝐸𝑃 𝑃

• 𝑄 can be represented as a local Markov network with graph 𝐺.

» Efficient CE Representation Lemma :

For every CE 𝑃 for a graphical game with graph 𝐺, there exists a CE 𝑄 for the game with the properties

• 𝑄 is a local Markov network with graph 𝐺

• 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

CE Representation Lemma

» CE conditions correspond to linear inequalities in the joint distribution values 𝑃(𝑎1, … , 𝑎𝑛)

» Distribution constraints also linear in 𝑃(𝑎1, … , 𝑎𝑛)

» 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 (𝑂(2𝑛)) by using linear programming (LP)

» Can we preserve succinctness of GG representation in CE computation?

Computation: Normal-form Games

» Variables: {𝑃𝑖( Ԧ𝑎𝑖)} (local neighborhood marginals)

» Global CE constraints (can be a large set)

• Best response: linear in {𝑃𝑖( Ԧ𝑎𝑖)}

• Global consistency: ensure {𝑃𝑖( Ԧ𝑎𝑖)} 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 𝑄 for tree graphical games with graph 𝐺 in time polynomial in the representation size of the game

» Can be extended to bounded tree-width graphical games

» 𝑄 is also a local Markov network with graph 𝐺.

» Can sample uniformly from set of CE in polynomial-time for bounded tree-width GG

Constraints

Linear Programming

» Variables: for every player I and assignment Ԧ𝑎𝑖, there is a variable 𝑃( Ԧ𝑎𝑖)

» LP Constraints:• CE Constraints: for all players 𝑖 and actions 𝑎, 𝑎′

𝑎𝑖:𝑎𝑖𝑖=𝑎

𝑃( Ԧ𝑎𝑖)𝑀𝑖( Ԧ𝑎𝑖) ≥

𝑎𝑖:𝑎𝑖𝑖=𝑎

𝑃 Ԧ𝑎𝑖 𝑀𝑖[ Ԧ𝑎𝑖[𝑖: 𝑎′]]

• Neighborhood Marginal Constraints: for all players 𝑖:∀ Ԧ𝑎𝑖: 𝑃 Ԧ𝑎𝑖 ≥ 0,

𝑎𝑖

𝑃 Ԧ𝑎𝑖 = 1

• Intersection Consistency Constraints: for all players 𝑖 and 𝑗, and for any assignment Ԧ𝑦𝑖𝑗 to the intersection 𝑁(𝑖) ∩ 𝑁(𝑗) :

𝑃 Ԧ𝑎𝑖𝑗 ≡

𝑎𝑖:𝑎𝑖𝑗=𝑦𝑖𝑗

𝑃 Ԧ𝑎𝑖 =

𝑎𝑗:𝑎𝑖𝑗=𝑦𝑖𝑗

𝑃𝑗 Ԧ𝑎𝑗 ≡ 𝑃𝑗( Ԧ𝑎𝑖𝑗)

Algorithm Result Lemma

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

» 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

CE in GG: Summary

» Efficient Algorithms for Exact Nash Computation in Trees.

» Strategy-Proof Algorithms for Distributed Nash Computation.

» Cooperative, Behavioral, and Other Equilibrium Notions.

Open Problems/Areas

Any questions?

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» [email protected]

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THANKS!

computational economics algorithmic game theory

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