Logarithmic Query Complexity for Approximate Nash Computation in Large Games
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Abstract
We investigate the problem of equilibrium computation for “large” nplayer games. Large games have a Lipschitztype property that no single player’s utility is greatly affected by any other individual player’s actions. In this paper, we mostly focus on the case where any change of strategy by a player causes other players’ payoffs to change by at most \(\frac {1}{n}\). We study algorithms having query access to the game’s payoff function, aiming to find εNash equilibria. We seek algorithms that obtain ε as small as possible, in time polynomial in n. Our main result is a randomised algorithm that achieves ε approaching \(\frac {1}{8}\) for 2strategy games in a completely uncoupled setting, where each player observes her own payoff to a query, and adjusts her behaviour independently of other players’ payoffs/actions. O(log n) rounds/queries are required. We also show how to obtain a slight improvement over \(\frac {1}{8}\), by introducing a small amount of communication between the players. Finally, we give extension of our results to large games with more than two strategies per player, and alternative largeness parameters.
Keywords
Game theory Lipschitz games Multiplayer games Uncoupled dynamics Equilibrium computation1 Introduction
In studying the computation of solutions of multiplayer games, we encounter the wellknown problem that a game’s payoff function has description length exponential in the number of players. One approach is to assume that the game comes from a conciselyrepresented class (for example, graphical games, anonymous games, or congestion games), and another one is to consider algorithms that have query access to the game’s payoff function.
In this paper, we study the computation of approximate Nash equilibria of multiplayer games having the feature that if a player changes her behaviour, she only has a small effect on the payoffs that result to any other player. These games, sometimes called large games, or Lipschitz games, have recently been studied in the literature, since they model various realworld economic interactions; for example, an individual’s choice of what items to buy may have a small effect on prices, where other individuals are not strongly affected. Note that these games do not have conciselyrepresented payoff functions, which makes them a natural class of games to consider from the querycomplexity perspective. It is already known how to compute approximate correlated equilibria for unrestricted nplayer games. Here we study the more demanding solution concept of approximate Nash equilibrium.
Large games (equivalently, smallinfluence games) are studied in Kalai [16] and Azrieli and Shmaya [1]. In these papers, the existence of pure εNash equilibria for \(\varepsilon = \gamma \sqrt {8n\log (2kn)}\) is established, where γ is the largeness/Lipschitz parameter of the game, and k is the number of pure strategies for each player. In particular, since we assume that \(\gamma = \frac {1}{n}\) and k = 2 we notice that ε = O(n^{− 1/2}) so that there exist arbitrarily accurate pure Nash equilibria in large games as the number of players increases. Kearns et al. [17] study this class of games from the mechanism design perspective of mediators who aim to achieve a good outcome to such a game via recommending actions to players.
Babichenko [2] studies large binaryaction anonymous games. Anonymity is exploited to create a randomised dynamic on pure strategy profiles that with high probability converges to a pure approximate equilibrium in O(n log n) steps.
Payoff query complexity has been recently studied as a measure of the difficulty of computing gametheoretic solutions, for various classes of games. Upper and lower bounds on query complexity have been obtained for bimatrix games [6, 7], congestion games [7], and anonymous games [11]. For general nplayer games (where the payoff function is exponential in n), the query complexity is exponential in n for exact Nash, also exact correlated equilibria [15]; likewise for approximate equilibria with deterministic algorithms (see also [4]). For randomised algorithms, query complexity is exponential for wellsupported approximate equilibria [3], which has since been strengthened to any εNash equilibria [5]. With randomised algorithms, the query complexity of approximate correlated equilibrium is Θ(log n) for any positive ε [10].
Our main result applies in the setting of completely uncoupled dynamics in equilibria computation. These dynamics have been studied extensively: Hart and MasColell [13] show that there exist finitememory uncoupled strategies that lead to pure Nash equilibria in every game where they exist. Also, there exist finite memory uncoupled strategies that lead to εNE in every game. Young’s interactive trial and error [18] outlines completely uncoupled strategies that lead to pure Nash equilibria with high probability when they exist. Regret testing from Foster and Young [8] and its nplayer extension by Germano and Lugosi in [9] show that there exist completely uncoupled strategies that lead to an εNash equilibrium with high probability. Randomisation is essential in all of these approaches, as Hart and MasColell [14] show that it is impossible to achieve convergence to Nash equilibria for all games if one is restricted to deterministic uncoupled strategies. This prior work is not concerned with rate of convergence; by contrast here we obtain efficient bounds on runtime. Convergence in adaptive dynamics for exact Nash equilibria is also studied by Hart and Mansour in [12] where they provide exponential lower bounds via communication complexity results. Babichenko [3] also proves an exponential lower bound on the rate of convergence of adaptive dynamics to an approximate Nash equilibrium for general binary games. Specifically, he proves that there is no kqueries dynamic that converges to an εWSNE in \(\frac {2^{{\Omega }(n)}}{k}\) steps with probability of at least 2^{−Ω(n)} in all nplayer binary games. Both of these results motivate the study of specific subclasses of these games, such as the “large” games studied here.
2 Preliminaries
We consider games with n players where each player has k actions \(\mathcal {A} = \{0, 1, ...,k1\}\). Let a = (a_{i}, a_{−i}) denote an action profile in which player i plays action a_{i} and the remaining players play action profile a_{−i}. We also consider mixed strategies, which are defined by the probability distributions over the action set \(\mathcal {A}\). We write p = (p_{i}, p_{−i}) to denote a mixedstrategy profile where p_{i} is a distribution over \(\mathcal {A}\) corresponding to the ith player’s mixed strategy. To be more precise, p_{i} is a vector \( (p_{ij})_{j = 1}^{k1}\) such that \({\sum }_{j = 1}^{k1}p_{ij} \leq 1\) where p_{ij} denotes the ith player’s probability mass on her jth strategy. Furthermore, we denote \(p_{i0} = 1  {\sum }_{j = 1}^{k1}p_{ij}\) to be the implicit probability mass the ith player places on her 0th pure strategy.
Each player i has a payoff function \(u_{i}\colon \mathcal {A}^{n} \rightarrow [0, 1]\) mapping an action profile to some value in [0, 1]. We will sometimes write \(u_{i}(p) = \mathbb {E}_{a\sim p}\left [u_{i}(a)\right ]\) to denote the expected payoff of player i under mixed strategy p. An action a is player i’s best response to mixed strategy profile p if \(a \in \text {argmax}_{j\in \mathcal {A}} u_{i}(j , p_{i})\).
We assume our algorithms or the players have no other prior knowledge of the game but can access payoff information through querying a payoff oracle\(\mathcal {Q}\). For each payoff query specified by an action profile \(a\in \mathcal {A}^{n}\), the query oracle will return \((u_{i}(a))_{i = 1}^{n}\), the ndimensional vector of payoffs to each player. Our goal is to compute an approximate Nash equilibrium with a small number of queries. In the completely uncoupled setting, a query works as follows: each player i chooses her own action a_{i} independently of the other players, and learns her own payoff u_{i}(a) but no other payoffs.
Definition 1 (Regret; (approximate) Nash equilibrium)
A mixed strategy profile p is an εapproximate Nash equilibrium (εNE) if for each player i, the regret satisfies reg(p, i) ≤ ε.
In Section 6.1 we will address the stronger notion of a wellsupported approximate Nash equilibrium. In essence, such an equilibrium is a mixedstrategy profile where players only place positive probability on actions that are approximately optimal. In order to precisely define this, we introduce \(supp(p_{i}) = \{j \in \mathcal {A} \  \ p_{ij} > 0 \}\) to be the set of actions that are played with positive probability in player i’s mixed strategy p_{i}.
Definition 2 (Wellsupported approximate Nash equilibrium)
An εWSNE is always an εNE, but the converse is not necessarily true as a player may place probability mass on strategies that are more than ε from optimal yet still maintain a low regret in the latter.
Observation 1
To find an exact Nash (or even, correlated) equilibrium of a large game,in the worst case it is necessary to query the game exhaustively, evenwith randomised algorithms. This uses a similar negative result for generalgames due to [15], and noting that we can obtain a strategically equivalentγlargegame (Definition3), by scaling down the payoffs into the interval [0, γ].
We will assume the following largeness condition in our games. Informally, such largeness condition implies that no single player has a large influence on any other player’s utility function.
Definition 3 (Large Games)
We will call γ the largeness parameter of the game; in [1] this quantity is called the Lipschitz value of the game. One immediate implication of the largeness assumption is the following Lipschitz property of the utility functions.
Lemma 1
For any playeri ∈ [n],and any action\(j\in \mathcal {A}\),the fixed utility functionu_{i}(j, p_{−i}) : [0, 1]^{(n− 1)×(k− 1)} → [0, 1] is aγLipschitzfunction of the second argumentp_{−i} ∈ [0, 1]^{(n− 1)×(k− 1)}w.r.t. theℓ_{1}norm.
Proof
Without loss of generality consider i = 1 and j = 0. Let q = p_{− 1} and \(q^{\prime } = p^{\prime }_{1}\) be two mixed strategy profiles for the other players. For i ≥ 2 and \(j \in \mathcal {A} \setminus \{0\}\), let \(\delta _{ij} = q^{\prime }_{ij}  q_{ij}\). Note that \(\q  q^{\prime }\_{1} = {\sum }_{ij} \delta _{ij}\).
Let e_{ij} be the unit vector that has a 1 in the (ij)th entry and 0 elsewhere. We first show that there exists an ordering of the discrete set {(ij)  2 ≤ i ≤ n, 1 ≤ j ≤ k} denoted by {α_{1}, α_{2}, ..., α_{(n− 1)(k− 1)}} such that for all ℓ = 1, ..., (n − 1)(k − 1), the vector \(q_{\ell } = q + {\sum }_{i = 1}^{\ell } \delta _{\alpha _{i}} e_{\alpha _{i}}\) represents valid mixed strategy profiles for players i ≥ 2.
Suppose that we fix i, and consider q_{i} and \(q^{\prime }_{i} \) as the mixed strategies of player i arising in q and q^{′}. We recall that these are vectors in [0, 1]^{k− 1} whose components sum is less than 1. We consider two cases. In the first, suppose that there exists a j such that δ_{ij} < 0 by definition, δ_{ij} < q_{ij}, hence q_{i} + δ_{ij}e_{j} is a valid mixed strategy for player i.
In the second, suppose that δ_{ij} > 0 for all j. Now suppose that \(\delta _{ij} > q_{i0} = 1  {\sum }_{j = 1}^{k1} q_{ij}\) for all j. If such is the case then \(q^{\prime }_{i}\) cannot possibly be a valid mixed strategy for player i, hence it must be the case that for some j, δ_{ij} < q_{i0}, hence once again q_{i} + δ_{ij}e_{j} is a valid mixed strategy for player i.
Since such a choice of valid updates by δ_{ij} can always be found for valid q_{i} and \(q^{\prime }_{i}\), we can recursively find valid shifts by δ_{ij} in a specific coordinate to reach \(q^{\prime }_{i}\) from q_{i}. If this is applied in order for all players i ≥ 2, the aforementioned claim holds and indeed \(q_{\ell } = q + {\sum }_{i = 1}^{\ell } \delta _{\alpha _{i}} e_{\alpha _{i}}\) for some ordering {α_{1}, ..., α_{(n− 1)(k− 1)}}.
From now on until Section 6 we will focus on \(\frac {1}{n}\)large binary action games where \(\mathcal {A} = \{0, 1\}\) and \(\gamma =\frac {1}{n}\). The reason for this is that the techniques we introduce can be more conveniently conveyed in the special case of \(\gamma =\frac {1}{n}\), and subsequently extended to general γ.
Recall that p_{i} denotes a mixed strategy of player i. In the special case of binaryaction games, we slightly abuse the notation to let p_{i} denote the probability that player i plays 1 (as opposed to 0), since in the binaryaction case, this single probability describes i’s mixed strategy.
The following notion of discrepancy will be useful.
Definition 4 (Discrepancy)
Estimating Payoffs for Mixed Profiles

For any input mixed strategy profile p, compute a new mixed strategy profile \(p^{\prime } = (1  \frac {\beta }{2})p + (\frac {\beta }{2})\mathbf {1}\) such that each player i is playing uniform distribution with probability \(\frac {\beta }{2}\) and playing distribution p_{i} with probability \(1  \frac {\beta }{2}\).

Let \(N = \frac {64}{\beta ^{3}} \log \left (8n /\delta \right )\), and sample N payoff queries randomly from p^{′}, and call the oracle \(\mathcal {Q}\) with each query as input to obtain a payoff vector.

Let \(\widehat u_{i,j}\) be the average sampled payoff to player i for playing action j.^{1} Output the payoff vector \((\widehat {u}_{ij})_{i\in [n], j\in \{0, 1\}}\).
Lemma 2
The lemma follows from Proposition 1 of [10] and the largeness property.
Extension to Stochastic Utilities
We consider a generalisation where the utility to player i of any pure profile a may consist of a probability distribution D_{a,i} over [0, 1], and if a is played, i receives a sample from D_{a,i}. The player wants to maximise her expected utility with respect to sampling from a (possibly mixed) profile, together with sampling from any D_{a,i} that results from a being chosen. If we extend the definition of \(\mathcal {Q}\) to output samples of the D_{a,i} for any queried profile a, then \(\mathcal {Q}_{\beta ,\delta }\) can be defined in a similar way as before, and simulated as above using samples from \(\mathcal {Q}\). Our algorithmic results extend to this setting.
3 Warmup: 0⋅25Approximate Equilibrium
In this section, we exhibit some simple procedures whose general approach is to query a constant number of mixed strategies (for which additive approximations to the payoffs can be obtained by sampling). Observation 2 notes that a \(\frac {1}{2}\)approximate Nash equilibrium can be found without using any payoff queries:
Observation 2
Consider the following “uniform” mixed strategy profile. Each player puts \(\frac {1}{2}\) probability mass on each action: for all i, \(p_{i} = \frac {1}{2}\) . Such a mixed strategy profile is a \(\frac {1}{2}\) approximate Nash equilibrium.
We present two algorithms that build on Observation 2 to obtain better approximations than \(\frac {1}{2}\). For simplicity of presentation, we assume that we have access to a mixed strategy query oracle \(\mathcal {Q}_{M}\) that returns exact expected payoff values for any input mixed strategy p. Our results continue to hold if we replace \(\mathcal {Q}_{M}\) by \(\mathcal {Q}_{\beta , \delta }\). ^{2}
Obtaining ε = 0⋅272

Let the players play the “uniform” mixed strategy. Call the oracle \(\mathcal {Q}_{M}\) to obtain the payoff values of u_{i}(0, p_{−i}) and u_{i}(1, p_{−i}) for each player i.

For each player i, if u_{i}(0, p_{−i}) − u_{i}(1, p_{−i}) > α, then set \(p_{i} = \frac {1}{2}  {\Delta }\); if u_{i}(1, p_{−i}) − u_{i}(0, p_{−i}) > α, set \(p_{i} = \frac {1}{2} + {\Delta }\); otherwise keep playing \(p_{i} = \frac {1}{2}\).
Theorem 1
If we use algorithmOneStepwith parameters\(\alpha = 2  \sqrt {\frac {11}{3}}\)and\({\Delta } = \sqrt {\frac {11}{48}}  \frac {1}{4}\),then the resulting mixed strategy profile is anεapproximateNash equilibrium withε ≤ 0⋅272.
Proof
Let p denote the “uniform” mixed strategy, and p^{′} denote the output strategy by OneStep. We know that ∥p − p^{′}∥_{1} ≤ nΔ. By Lemma 1, we know that for any player i and action j, \(u_{i}(j, p_{i})  u_{i}(j, p_{j}^{\prime })\leq {\Delta }\).
Obtaining ε = 0⋅25

Start with the “uniform” mixed strategy profile, and query the oracle \(\mathcal {Q}_{M}\) for the payoff values. Let b_{i} be player i’s best response.

For each player i, set the probability of playing their best response b_{i} to be \(\frac {3}{4}\). Call \(\mathcal {Q}_{M}\) to obtain payoff values for this mixed strategy profile, and let \(b^{\prime }_{i}\) be each player i’s best response in the new profile.

For each player i, if \(b_{i}\neq b^{\prime }_{i}\), then resume playing \(p_{i} = \frac {1}{2}\). Otherwise maintain the same mixed strategy from the previous step.
Theorem 2
The mixed strategy profile output byTwoStepis anεapproximateNash equilibrium withε ≤ 0⋅25.
Proof
Let p denote the “uniform” strategy profile, p^{′} denote the strategy profile after the first adjustment, and p^{″} denote the output strategy profile by TwoStep.
 1.
The discrepancy \(disc(p, i) > \frac {1}{2}\);
 2.
The discrepancy \(disc(p, i) \leq \frac {1}{2}\) and player i returns to the uniform mixed strategy at the end;
 3.
The discrepancy \(disc(p, i) \leq \frac {1}{2}\) and player i does not return to the uniform mixed strategy in the end.
Therefore, we know \(disc(p^{\prime \prime },i) \leq \frac {1}{2}\), and hence the regret \(reg(p^{\prime \prime },i) \leq \frac {1}{4}\).
If in the end her best response changes to 0, then the regret is bounded by \(reg(p^{\prime \prime },i) \leq \frac {1}{8}\). Otherwise if the best response remains to be 1, then the regret is again bounded by \(reg(p^{\prime \prime }, i) \leq \frac {1}{4}\)
Hence, in all of the cases above we could bound the player’s regret by \(\frac {1}{4}\). □
4 \(\frac {1}{8}\)Approximate Equilibrium via Uncoupled Dynamics
In this section, we present our main algorithm that achieves approximate equilibria with \(\varepsilon \approx \frac {1}{8}\) in a completely uncoupled setting. In order to arrive at this we first model game dynamics as an uncoupled continuoustime dynamical system where a player’s strategy profile updates depend only on her own mixed strategy and payoffs. Afterwards we present a discretetime approximation to these continuous dynamics to arrive at a querybased algorithm for computing \((\frac {1}{8} + \alpha )\)Nash equilibrium with query complexity logarithmic in the number of players. Here, α > 0 is a parameter that can be chosen, and the number of mixedstrategy profiles that need to be tested is inversely proportional to α. Finally, as mentioned in Section 2, we recall that these algorithms carry over to games with stochastic utilities, for which we can show that our algorithm uses an essentially optimal number of queries.
Throughout the section, we will rely on the following notion of a strategy/payoff state, capturing the information available to a player at any moment of time.
Definition 5 (Strategypayoff state)
For any player i, the strategy/payoff state for player i is defined as the ordered triple s_{i} = (v_{i1}, v_{i0}, p_{i}) ∈ [0, 1]^{3}, where v_{i1} and v_{i0} are the player’s utilities for playing pure actions 1 and 0 respectively, and p_{i} denotes the player’s probability of playing action 1. Furthermore, we denote the player’s discrepancy by D_{i} = v_{i1} − v_{i0} and we let \(p_{i}^{*}\) denote the probability mass on the best response, that is if v_{i1} ≥ v_{i0}, \(p_{i}^{*} = p_{i}\), otherwise \(p_{i}^{*} = 1p_{i}\).
4.1 ContinuousTime Dynamics
First, we will model game dynamics in continuous time, and assume that a player’s strategy/payoff state (and thus all variables it contains) is a differentiable timevalued function. When we specify these values at a specific time t, we will write s_{i}(t) = (v_{i1}(t), v_{i0}(t), p_{i}(t)). Furthermore, for any timedifferentiable function g, we denote its time derivative by \(\dot {g} = \frac {d}{dt} g\). We will consider continuous game dynamics formally defined as follows.
Definition 6 (Continuous game dynamic)
A continuous game dynamic consists of an update function f that specifies a player’s strategy update at time t. Furthermore, f depends only on s_{i}(t) and \(\dot {s}_{i}(t)\). In other words, \(\dot {p}_{i}(t) = f(s_{i}(t), \dot {s}_{i}(t))\) for all t.
Observation 3
We note that in this framework, a specific player’s updates do not dependon other players’ strategy/payoff states nor their history of play. This willeventually lead us to uncoupled Nash equilibria computation in Section 4.2.
Lemma 3
If player i’s strategy/payoff state satisfies \(s_{i}\in \mathcal {P}\) , then her regret is at most \(\frac {1}{8}\) .
Proof
This follows from the fact that a player’s regret can be expressed as \(D_{i}(1p_{i}^{*})\) and the fact that all points on \(\mathcal {P}\) also satisfy \(p_{i}^{*} = \frac {1}{2}(1+D_{i})\). In particular, the maximal regret of \(\frac {1}{8}\) is achieved when \(D_{i} = \frac {1}{2}\) and \(p_{i}^{*} = \frac {3}{4}\). □
Next, we want to show there exists a dynamic that allows all players to eventually reach \(\mathcal {P}\) and remain on it over time. We notice that for a specific player, \(\dot {v}_{i1}\), \(\dot {v}_{i0}\) and subsequently \(\dot {D}_{i}\) measure the cumulative effect of other players shifting their strategies. However, if we limit how much any individual player can change their mixed strategy over time by imposing \(\dot {p}_{i} \leq 1\) for all i, Lemma 1 guarantees \(\dot {v}_{ij} \leq 1\) for j = 0, 1 and consequently \(\dot {D}_{i} \leq 2\). With these quantities bounded, we can consider an adversarial framework where we construct game dynamics by solely assuming that \(\dot {p}_{i}(t) \leq 1\), \(\dot {v}_{ij}(t) \leq 1\) for j = 0, 1 and \(\dot {D}_{i}(t) \leq 2\) for all times t ≥ 0.
Now assume an adversary controls \(\dot {v}_{i0}\), \(\dot {v}_{i1}\) and hence \(\dot {D}_{i}\), one can show that if a player sets \(\dot {p}_{i}(t) = \frac {1}{2}(\dot {v}_{i1}(t)  \dot {v}_{i0}(t))\), then she could stay on \(\mathcal {P}\) whenever she reaches the subspace.
Lemma 4
If \(s_{i}(0) \in \mathcal {P}\) , and \(\dot {p}_{i}(t) = \frac {1}{2}(\dot {v}_{i1}(t)  \dot {v}_{i0}(t))\) , then \(s_{i}(t) \in \mathcal {P} \ \forall \ t \geq 0\) .
Theorem 3
Proof
From Lemma 4 it is clear that once a player reaches \(\mathcal {P}\) they never leave the plane. It remains to show that it takes at most \(\frac {1}{2}\) time units to reach \(\mathcal {P}\).
Since \(p_{i}(0) = p_{i}^{*}(0) = \frac {1}{2}\), it follows that if \(s_{i}(0) \notin \mathcal {P}\) then \(p_{i}^{*}(0) < \frac {1}{2}(1 + D_{i}(0))\). On the other hand, if we assume that \(\dot {p}_{i}^{*}(t) = 1\) for \(t \in [0,\frac {1}{2}]\), and that player preferences do not change, then it follows that \(p_{i}^{*}(\frac {1}{2}) = 1\) and \(p_{i}^{*}(\frac {1}{2}) \geq \frac {1}{2}(1 + D_{i}(\frac {1}{2}))\), where equality holds only if \(D_{i}(\frac {1}{2}) = 1\). By continuity of \(p_{i}^{*}(t)\) and D_{i}(t) it follows that for some \(k \leq \frac {1}{2}\), \(s_{i}(k) \in \mathcal {P}\). It is simple to see that the same holds in the case where preferences change. □
4.2 Discrete TimeStep Approximation
The continuoustime dynamics of the previous section hinge on obtaining expected payoffs in mixed strategy profiles, thus we will approximate expected payoffs via \(\mathcal {Q}_{\beta ,\delta }\). Our algorithm will have each player adjusting their mixed strategy over rounds, and in each round query \(\mathcal {Q}_{\beta , \delta }\) to obtain the payoff values.
Since we are considering discrete approximations to UCN, the dynamics will no longer guarantee that strategy/payoff states stay on the plane \(\mathcal {P}\). For this reason we define the following region around \(\mathcal {P}\):
Definition 7
Let \(\mathcal {P}^{\lambda } = \{ s_{i} \  \ s_{i} \cdot \vec {n} \in [\frac {1}{2}  \lambda , \frac {1}{2} + \lambda ]\}\), with normal vector \(\vec {n} = (\frac {1}{2},\frac {1}{2},1)\). Equivalently, \(\mathcal {P}^{\lambda } = \{s_{i} \  \ p_{i}^{*} = \frac {1}{2}(1+D_{i}) + c, \ c \in [\lambda ,\lambda ]\}\).
Just as in the proof of Lemma 3, we can use the fact that a player’s regret is \(D_{i}(1p_{i}^{*})\) to bound regret on \(\mathcal {P}^{\lambda }\).
Lemma 5
The worst case regret of any strategy/payoff state in \(\mathcal {P}^{\lambda }\) is \(\frac {1}{8}(1 + 2\lambda )^{2}\) . This is attained on the boundary: \(\partial \mathcal {P}^{\lambda } = \{ s_{i} \  \ s_{i} \cdot \vec {n} = \frac {1}{2} \pm \lambda \}\) .
Corollary 1
For a fixedα > 0,if\(\lambda = \frac {\sqrt {1 + 8\alpha }  1}{2}\),then\(\mathcal {P}^{\lambda }\)attains a maximal regret of\(\frac {1}{8} +\alpha \).
We present an algorithm in the completely uncoupled setting, UN(α, η), that for any parameters α, η ∈ (0, 1] computes a \((\frac {1}{8} + \alpha )\)Nash equilibrium with probability at least 1 − η.
Since p_{i}(t) ∈ [0, 1] is the mixed strategy of the ith player at round t we let \(p(t) = (p_{i}(t))_{i = 1}^{n}\) be the resulting mixed strategy profile of all players at round t. Furthermore, we use the mixed strategy oracle \(\mathcal {Q}_{\beta ,\delta }\) from Lemma 2 that for a given mixed strategy profile p returns the vector of expected payoffs for all players with an additive error of β and a correctness probability of 1 − δ.
The following lemma is used to prove the correctness of UN(α, η):
Lemma 6
Suppose that\(w \in \mathbb {R}^{3}\)with∥w∥_{∞} ≤ λand let function\(h(x) = x \cdot \vec {n}\),where\(\vec {n}\)is the normal vector of\(\mathcal {P}\).Thenh(x + w) − h(x) ∈ [− 2λ, 2λ].Furthermore, ifw_{3} = 0,thenh(x + w) − h(x) ∈ [−λ, λ].
Proof
Theorem 4
With probability1 − η,UN(α, η)correctly returns a\((\frac {1}{8} + \alpha )\)approximateNash equilibrium by using\(O(\frac {1}{\alpha ^{4}} \log \left (\frac {n}{\alpha \eta } \right ) ) \)queries.
Proof
By Lemma 2 and union bound, we can guarantee that with probability at least 1 − η all sample approximations to mixed payoff queries have an additive error of at most \({\Delta } = \frac {\lambda }{4}\). We will condition on this accuracy guarantee in the remainder of our argument. Now we can show that for each player there will be some round k ≤ N, such that at the beginning of the round their strategy/payoff state lies in \(\mathcal {P}^{\lambda /2}\). Furthermore, at the beginning of all subsequent rounds t ≥ k, it will also be the case that their strategy/payoff state lies in \(\mathcal {P}^{\lambda /2}\).
The reason any player generally reaches \(\mathcal {P}^{\lambda /2}\) follows from the fact that in the worst case, after increasing p^{∗} by Δ for N rounds, p^{∗} = 1, in which case a player is certainly in \(\mathcal {P}^{\lambda /2}\). Furthermore, Lemma 6 guarantees that each time p^{∗} is increased by Δ, the value of \(\widehat {s}_{i} \cdot \vec {n}\) changes by at most \(\frac {\lambda }{2}\) which is why \(\widehat {s}_{i}\) are always steered towards \(\mathcal {P}^{\lambda /4}\). Due to inherent noise in sampling, players may at times find that \(\widehat {s}_{i}\) slightly exit \(\mathcal {P}^{\lambda /4}\) but since additive errors are at most \(\frac {\lambda }{4}\). We are still guaranteed that true s_{i} lie in \(\mathcal {P}^{\lambda /2}\).
The second half of step 4 forces a player to remain in \(\mathcal {P}^{\lambda /2}\) at the beginning of any subsequent round t ≥ k. The argumentation for this is identical to that of Lemma 4 in the continuous case.
It is not difficult to see that \(\frac {1}{\lambda } = O(\frac {1}{\alpha })\) which implies that the number of queries made is \(O \left (\frac {1}{\alpha ^{4}} \log \left (\frac {n}{ \alpha \eta } \right ) \right )\) in the limit. □
4.3 Logarithmic Lower Bound
As mentioned in the preliminaries section, all of our previous results extend to stochastic utilities. In particular, if we assume that G is a game with stochastic utilities where expected payoffs are large with parameter \(\frac {1}{n}\), then we can apply UN(α, η) with O(log(n)) queries to obtain a mixed strategy profile where no player has more than \(\frac {1}{8} + \alpha \) incentive to deviate. Most importantly, for ℓ > 2, we can use the same methods as [10] to lower bound the query complexity of computing a mixed strategy profile where no player has more than \((\frac {1}{2}  \frac {1}{\ell })\) incentive to deviate.
Theorem 5
Ifℓ > 2,the query complexity of computing a mixedstrategy profile where no player has more than\((\frac {1}{2}  \frac {1}{\ell })\)incentive to deviate for stochastic utility games is Ω(log ℓ(ℓ− 1)(n)).Alongside Theorem 4 this implies the query complexity ofcomputing mixed strategy profiles where no player has more than\(\frac {1}{8}\)incentive to deviate in stochastic utility games isΘ(log(n)).
Proof
Suppose that we have n players and that ℓ > 2. For every b ∈{0, 1}^{n} we can construct a stochastic utility game G_{b} as follows: For each player i, the utility of strategy b_{i} is bernoulli with bias \(\frac {\ell }{\ell 1}\) and the utility of strategy 1 − b_{i} is bernoulli with bias \(\frac {1}{\ell }\). Note that this game is trivially \(\left (\frac {1}{n} \right )\)Lipschitz, as each player’s payoff distributions are completely independent of other players’ strategies. □
Suppose that \(\mathcal {G}\) is the uniform distribution on the set of all G_{b}, then using the same argumentation as Theorem 3 of [10], we get the following:
Theorem 6
Let\(\mathcal {A}\)be a deterministic payoffquery algorithm that uses at most log ℓ(ℓ− 1)(n) queries and outputs a mixed strategyp. If\(\mathcal {A}\)performs on\(\mathcal {G}\),then with probability more than\(\frac {1}{2}\),there will exist a player with a regret greater than\(\frac {1}{2}  \frac {1}{\ell }\)inp.
We can immediately apply Yao’s minimax principle to this result to complete the proof.
5 Achieving \(\varepsilon < \frac {1}{8}\) with Communication
We return to continuous dynamics to show that we can obtain a worstcase regret of slightly less than \(\frac {1}{8}\) by using limited communication between players, thus breaking the uncoupled setting we have been studying until now.
First of all, let us suppose that initially \(p_{i}(0) = \frac {1}{2}\) for each player i and that UCN is run for \(\frac {1}{2}\) time units so that strategy/payoff states for each player lie on \(\mathcal {P} = \{ s_{i} \  \ p_{i}^{*} = \frac {1}{2}(1 + D_{i}) \}\). We recall from Lemma 3 that the worst case regret of \(\frac {1}{8}\) on this plane is achieved when \(p_{i}^{*} = \frac {3}{4}\) and \(D_{i} = \frac {1}{2}\). We say a player is bad if they achieve a regret of at least 0⋅12, which on \(\mathcal {P}\) corresponds to having \(p_{i}^{*} \in [0{\cdot }7,0{\cdot }8]\). Similarly, all other players are good. We denote 𝜃 ∈ [0, 1] as the proportion of players that are bad. Furthermore, as the following lemma shows, we can in a certain sense assume that \(\theta \leq \frac {1}{2}\).
Lemma 7
If\(\theta > \frac {1}{2}\),then for a period of 0⋅15 time units, we can allow each bad player to shift to their best response withunit speed, and have all good players update according toUCNto stay on\(\mathcal {P}\).After this movement, at most1 − 𝜃players are bad.
Proof
If i is a bad player, in the worst case scenario, \(\dot {D}_{i} = 2\), which keeps their strategy/payoff state, s_{i}, on the plane \(\mathcal {P}\). However, at the end of 0⋅15 time units, they will have \(p_{i}^{*} > 0{\cdot }85\), hence they will no longer be bad. On the other hand, since the good players follow the dynamic, they stay on \(\mathcal {P}\), and at worst, all of them become bad. □
Observation 4
After this movement, players who werebad are the only players possibly away from\(\mathcal {P}\)and they have a discrepancy that is greater than 0 ⋅ 1.Furthermore, all players who become bad lie on\(\mathcal {P}\).
 1.
Have all players begin with \(p_{i}(0) = \frac {1}{2}\)
 2.
Run UCN for \(\frac {1}{2}\) time units.
 3.
Measure, 𝜃, the proportion of bad players. If \(\theta > \frac {1}{2}\) apply the dynamics of Lemma 7.
 4.
Let all bad players use \(\dot {p}_{i}^{*} = 1\) for \({\Delta } = \frac {1}{220}\) time units.
Theorem 7
If all players follow the aforementioned dynamic, no single player will have a regret greater than \(\frac {1}{8}  \frac {1}{220}\) .
In essence one shows that if Δ is a small enough time interval (less than 0⋅1 to be exact), then all bad players will unilaterally decrease their regret by at least 0⋅1Δ and good players won’t increase their regret by more than Δ. The time step \({\Delta } = \frac {1}{220}\) is thus chosen optimally.
Proof
We have seen via Lemma 7 that after step 3 the proportion of bad players is at most \(\theta \leq \frac {1}{2}\), we wish to show that step 4 reduces maximal regret by at least \(\frac {1}{220}\) for every bad player whilst maintaining a low regret for good players.
As a final note, we see that this process requires one round of communication in being able to perform the operations in Lemma 7, that is we need to know if \(\theta > \frac {1}{2}\) or not to balance player profiles so that there are at most the same number of bad players to good players. Furthermore, in exactly the same fashion as UN(α, η), we can discretise the above process to obtain a querybased algorithm that obtains a regret of \(\frac {1}{8}  \frac {1}{220} + \alpha < \frac {1}{8}\) for arbitrary α.
6 Extensions

(Section 6.1) We extend the algorithm UCN to large games with a more general largeness parameter \(\gamma = \frac {c}{n} \in [0, 1]\), where c is a constant.

(Section 6.2) We consider large games with k actions and largeness parameter \(\frac {c}{n}\) (previously we focused on k = 2). Our algorithm used a new uncoupled approach that is substantially different from the previous ones we have presented.
6.1 Continuous Dynamics for Binaryaction Games with Arbitrary γ
Theorem 8
Notice that unlike UCN, this dynamic is no longer necessarily a continuously differentiable function with respect to time when c > 1. However, it is still continuous.
Theorem 9
Suppose that\(\gamma = \frac {c}{n}\)and that a player’s strategy/payoff state lies on\(\mathcal {P}_{\gamma }\),then her regret is at most\(\frac {c}{8}\)forc ≤ 2and her regret is at most\(\frac {1}{2}  \frac {1}{2c}\)forc > 2.Furthermore, the equilibria obtained are alsocWSNE.
Proof
If c ≤ 2, then regret is maximised when \(D = \frac {c}{2}\) and consequently when \(p^{*} = \frac {3}{4}\). This results in a regret of \(\frac {c}{8}\). On the other hand, if c > 2, then regret is maximised when D = 1 and consequently \(p^{*} = \frac {1}{2} + \frac {1}{2c}\). This results in a regret of \(\frac {1}{2}  \frac {1}{2c}\).
As for the second part of the theorem, from the definition of \(\mathcal {P}_{\gamma }\) and from the definition of εWSNE in Section 2 it is straightforward to see that when D ≥ c, p^{∗} = 1 which means that no weight is put on the strategy whose utility is at most c from that of the best response. □
Thus we obtain a regret that is better than simply randomising between both strategies, although as should be expected, the advantage goes to zero as the largeness parameter increases.
6.1.1 Discretisation and Query Complexity
In the same way as UN(α, η), where we discretised UN, Theorem 7 can be discretised to yield the following result.
Theorem 10
6.2 Equilibrium Computation for kaction Games
When the number of pure strategies per player is k > 2, the initial “strawman” idea corresponding to Observation 2 is to have all n players randomise uniformly over their k strategies. Notice that the resulting regret may in general be as high as \(1\frac {1}{k}\). In this section we give a new uncoupleddynamics approach for computing approximate equilibria in kaction games where (for largeness parameter \(\gamma =\frac {1}{n}\)) the worstcase regret approaches \(\frac {3}{4}\) as k increases, hence improving over uniform randomisation over all strategies. Recall that in general we are considering \(\gamma = \frac {c}{n}\) for fixed c ∈ [0, n]. The following is just a simple extension of the payoff oracle \(\mathcal {Q}_{\beta ,\delta }\) to the setting with k actions: for any input mixed strategy profile p, the oracle will with probability at least 1 − δ, output payoff estimates for p with error at most β for all n players.
Estimating Payoffs for Mixed Profiles in kaction Games

For any input mixed strategy profile p, compute a new mixed strategy profile \(p^{\prime } = (1  \frac {\beta }{2})p + (\frac {\beta }{2k})\mathbf {1}\) such that each player i is playing uniform distribution with probability \(\frac {\beta }{2}\) and playing distribution p_{i} with probability \(1  \frac {\beta }{2}\).

Let \(m = \frac {64k^{2}}{\beta ^{3}} \log \left (8n /\delta \right )\), and sample m payoff queries randomly from p^{′}, and call the oracle \(\mathcal {Q}\) with each query as input to obtain a payoff vector.

Let \(\widehat u_{i,j}\) be the average sampled payoff to player i for playing action j.^{3} Output the payoff vector \((\widehat {u}_{ij})_{i\in [n], j\in \{0, 1\}}\).
As in previous sections, we begin by assuming that our algorithm has access to \(\mathcal {Q}_{M}\), the more powerful query oracle that returns exact expected payoffs with regards to mixed strategies. We will eventually show in Section 6.2.1 that this does not result in a loss of generality, as when utilising \(\mathcal {Q}_{\beta , \delta }\) we incur a bounded additive loss with regards to the approximate equilibria we obtain.
The general idea of the analysis of Algorithm 2 is the following. In each time step, a player’s utilities change by at most nγ/N = c/N. Hence, at the completion of Algorithm 2, block N is allocated to a nearlyoptimal strategy, and generally, block N − r is allocated to a strategy whose closeness to optimality goes down as r increases, but enables us to derive the improved overall performance of each player’s mixed strategy.
Theorem 11
Notice for example that for \(\gamma =\frac {1}{n}\) (i.e. putting c = 1), each player’s regret is at most \(\frac {3}{4}+\frac {1}{2N}\), so we can make this arbitrarily close to \(\frac {3}{4}\) since N is a parameter of the algorithm.
Proof
For an arbitrary player i ∈ [n], in each step t = 1, ..., N, probability block B_{t} is reassigned to i’s current best response.
In fact, we can slightly improve the bounds in Theorem 9 via introducing a dependence on k. In order to do so, we need to introduce some definitions first.
Definition 8
Definition 9
The next important question is what valid assignment of blocks to regret values results in the maximal amount of total regret for a player. In Fig. 4, Block 1 is assigned to strategy 1, Blocks 2,3, and 7 are assigned to strategy 2, blocks 4 and 5 are assigned to strategy 3, block 5 is assigned to strategy 4 and finally blocks 8 and 9 are assigned to strategy 5.
Theorem 12
For any fixedR_{1}, ..., R_{k},the worst case assignment of probability blocksB_{b}to strategies corresponds to a left sum of\(\mathcal {A}^{(1+\frac {1}{N}), 2c}\)for some partition of\([0, 1+\frac {1}{N}]\)with cardinality at mostk − 1.
This previous theorem reduces the problem of computing worst case regret to that of computing maximal left sums under arbitrary partitions. To that end, we define the precise worstcase partition value we will be interested in.
Definition 10
We can explicity compute these values which in turn will bound a player’s maximal regret.
Lemma 8
\(\mathcal {A}^{1,1}_{k} = \left (\frac {1}{2} \right ) \left (\frac {k}{k + 1} \right )\) which is obtained on the partition \(\mathcal {P} = \{\frac {1}{k + 1}, \frac {2}{k + 1}, ...,\frac {k}{k + 1}\}\)
Proof
This result follows from induction and selfsimilarity of the original triangle. For k = 1, our partitions consist of a single point x ∈ [0, 1] hence the area under the triangle will be \(\mathcal {A}^{1,1}_{1}(x) = (1x)x\) which as a quadratic function of x has a maximum at \(x = \frac {1}{2}\). At this point we get \(\mathcal {A}^{1,1}_{1}(x) = \frac {1}{2} \cdot \frac {1}{2}\) as desired.
It is straightforward to see that \(\mathcal {A}^{\prime }(x)\) is maximised when \(x = \frac {1}{k + 2}\). Consequently the maximal truncated area arises from the partition where \(x_{i} = \frac {i}{n + 2}\) which in turn proves our claim. □
Via linear scaling, one can extend the above result to arbitrary base and height values b, h.
Corollary 2
Forbh ≤ 1,\(\mathcal {A}^{b,h}_{k} = \left (\frac {bh}{2} \right ) \left (\frac {k}{k + 1} \right )\)which is obtained on the partition\(\mathcal {P} = \{\frac {b}{k + 1}, \frac {2b}{k + 1}, ...,\frac {kb}{k + 1}\}\)
Corollary 3
Proof
For the first case (when \(\frac {k}{k + 1} \leq \frac {b}{h}\)), let us consider \(\mathcal {B}^{b,h}\) to be the the triangle with base b and height h that unlike \(\mathcal {A}^{b,h}\) is not truncated at unit height. From scaling our previous result from Corollary 2, the largest kelement left sum for \(\mathcal {B}^{b,h}\) occurs for the partition \(\mathcal {P} = \{\frac {b}{k + 1}, \frac {2b}{k + 1}, ...,\frac {bk}{k + 1} \}\). However, from the fact that \(\mathcal {A}^{b,h} \subset \mathcal {B}^{b,h}\), at precisely these values the left sums of \(\mathcal {P}\) for both geometric figures coincide. It follows that this partition also gives a maximal kelement partition for left sums of \(\mathcal {A}^{b,h}\) and thus the claim holds.
On the other hand, let us know consider the case where \(\frac {k}{k + 1} > \frac {b}{h}\). In a similar spirit to previous proofs, let us define \(\mathcal {A}(x): [0,b] \rightarrow \mathbb {R}\) to be the maximal leftsum under \(\mathcal {A}^{b,h}\) for a given partition \(\mathcal {P}\) whose rightmost element is x. From Figs. 4 and 5, it should be clear that we should only consider \(x \in [0,\frac {b}{h}]\), because if ever we have a \(x \geq \frac {b}{h}\), that would correspond to some block being assigned a regret value of R_{j} = 1 for some strategy j. However with the existence of such a maximal regret strategy, the greedy allotment of blocks to strategies would assign the most blocks possible to strategy j (or some other maximal regret strategy), which would correspond again to the final element in our partition being \(\frac {b}{h}\).
Finally, we can combine everything above to obtain:
Theorem 13
Proof
This just a straightforward application of Theorem 10 and Corollaries 2 and 3. □
6.2.1 Query Complexity of Block Method
In the above analysis we assumed access to a mixed strategy oracle as we computed expected payoffs at each timestep for all players. When using \(\mathcal {Q}_{\beta , \delta }\) however, there is an additive error and a bounded correctness probability to take into account.
Theorem 14
InBU, if queries incorporate an additive error ofαon expected utilities, for any fixed choice ofR_{1}, ..., R_{k},the worst case assignment of probability blocksB_{b}to strategies corresponds to a left sum of\(\mathcal {A}^{(1+\frac {1}{N} + \frac {\beta }{2c}), 2c}\)for some partition of\([0, 1+\frac {1}{N}]\)with cardinality at mostk − 1.
Finally, since our approximate query oracle is correct with a bounded probability, in order to assure that the same additive error of β holds on all N queries of BU, we need to impose a correctness probability of \(\frac {\delta }{N}\) in order to achieve the former with a union bound. This leads to the following query complexity result for BU.
Theorem 15
Once again, it is interesting to note that the first regret bounds we derived do not depend on k. It is also important to note the regret has an extra term of the form \(O(\frac {1}{N})\) in the number of probability blocks. Although this can be minimised in the limit, there is a price to be paid in query complexity, as this would involve a larger number of rounds in the computation of approximate equilibria.
6.3 Comparison Between Both Methods
c ≤ 1  1 ≤ c ≤ 2  c ≥ 2  

UNC  \(\frac {c}{8}\)  \(\frac {c}{8}\)  \(\frac {1}{2}  \frac {1}{2c}\) 
BU  \(\frac {c}{2}\)  \(1  \frac {1}{2c}\)  \(1  \frac {1}{2c}\) 
One can see that UNC does better by a multiplicative factor of \(\frac {1}{4}\) in the case of small c and better by an additive factor of \(\frac {1}{2}\) for large c.
7 Conclusion and Further Research
The obvious question raised by our results is the possible improvement in the additive approximation obtainable. Since pure approximate equilibria are known to exist for these games, the search for such equilibria is of interest. A slightly weaker objective (but still stronger than the solutions we obtain here) is the search for wellsupported approximate equilibria in cases where c > 1 and for better wellsupported approximate equilibria in general.
There is also the question of lower bounds, especially in the completely uncoupled setting. Our algorithms are randomised (estimating the payoffs that result from a mixed strategy profile via random sampling) and one might also ask what can be achieved using deterministic algorithms.
Footnotes
 1.
If the player i never plays an action j in any query, set \(\widehat u_{i,j} = 0\).
 2.
In particular, if we use \(\mathcal {Q}_{\beta ,\delta }\) for our query access, then with probability at least 1 − δ we will get (ε + O(β))approximate equilibrium, where ε is the approximation performance obtainable via access to \(\mathcal {Q}_{M}\).
 3.
If the player i never plays an action j in any query, set \(\widehat u_{i,j} = 0\).
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