Logarithmic Query Complexity for Approximate Nash Computation in Large Games

In studying the computation of solutions of multi-player games, we encounter the well-known 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 concisely-represented 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 multi-player 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 real-world 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 concisely-represented payoff functions, which makes them a natural class of games to consider from the query-complexity perspective. It is already known how to compute approximate correlated equilibria for unrestricted n-player games. Here we study the more demanding solution concept of approximate Nash equilibrium.

Large games (equivalently, small-influence games) are studied in Kalai [16] and Azrieli and Shmaya [1]. In these papers, the existence of pure ε-Nash 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 binary-action 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 game-theoretic 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 n-player 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 well-supported approximate equilibria [3], which has since been strengthened to any ε-Nash 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 Mas-Colell [13] show that there exist finite-memory uncoupled strategies that lead to pure Nash equilibria in every game where they exist. Also, there exist finite memory uncoupled strategies that lead to ε-NE 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 n-player extension by Germano and Lugosi in [9] show that there exist completely uncoupled strategies that lead to an ε-Nash equilibrium with high probability. Randomisation is essential in all of these approaches, as Hart and Mas-Colell [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 k-queries dynamic that converges to an ε-WSNE in \(\frac {2^{{\Omega }(n)}}{k}\) steps with probability of at least 2^−Ω(n) in all n-player binary games. Both of these results motivate the study of specific subclasses of these games, such as the “large” games studied here.

We consider games with n players where each player has k actions \(\mathcal {A} = \{0, 1, ...,k-1\}\). 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 mixed-strategy profile where p_i is a distribution over \(\mathcal {A}\) corresponding to the i-th player’s mixed strategy. To be more precise, p_i is a vector \( (p_{ij})_{j = 1}^{k-1}\) such that \({\sum }_{j = 1}^{k-1}p_{ij} \leq 1\) where p_ij denotes the i-th player’s probability mass on her j-th strategy. Furthermore, we denote \(p_{i0} = 1 - {\sum }_{j = 1}^{k-1}p_{ij}\) to be the implicit probability mass the i-th player places on her 0-th 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 n-dimensional 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.

In Section 6.1 we will address the stronger notion of a well-supported approximate Nash equilibrium. In essence, such an equilibrium is a mixed-strategy 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.

An ε-WSNE is always an ε-NE, 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.

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.

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.

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 binary-action 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 binary-action case, this single probability describes i’s mixed strategy.

The following notion of discrepancy will be useful.

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:

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

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 continuous-time dynamical system where a player’s strategy profile updates depend only on her own mixed strategy and payoffs. Afterwards we present a discrete-time approximation to these continuous dynamics to arrive at a query-based 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 mixed-strategy 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.

4.1 Continuous-Time 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 time-valued 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 time-differentiable 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.

A central object of interest in our continuous dynamic is a linear sub-space \(\mathcal {P} \subset [0, 1]^{3}\) such that all strategy/payoff states in it incur a bounded regret. Formally, we will define \(\mathcal {P}\) via its normal vector \(\vec {n} = (-\frac {1}{2},\frac {1}{2},1)\) so that \(\mathcal {P} = \{ s_{i} | \ s_{i} \cdot \vec {n} = \frac {1}{2} \}\). Equivalently, we could also write \(\mathcal {P} = \{s_{i} \ | \ p_{i}^{*} = \frac {1}{2}(1+D_{i})\}\). (See Fig. 1 for a visualisation.) With this observation, it is straightforward to see that any player with strategy/payoff state in \(\mathcal {P}\) has regret at most \(\frac {1}{8}\).

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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 \(D_{i}(1-p_{i}^{*})\) 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

Under the initial conditions \(p_{i}(0) = \frac {1}{2}\) for all i, the following continuous dynamic, Uncoupled Continuous Nash (UCN) , has all players reach \(\mathcal {P}\) in at most \(\frac {1}{2}\) time units. Furthermore, upon reaching \(\mathcal {P}\) a player never leaves.

$$\dot{p}_{i}(t) = f(s_{i}(t), \dot{s}_{i}(t)) = \left\{\begin{array}{cl} ~~~1 & \text{ if } s_{i} \notin \mathcal{P} \text{ and } v_{i1} \geq v_{i0} \\ -1 & \text{ if } s_{i} \notin \mathcal{P} \text{ and } v_{i1} < v_{i0} \\ \hfill \frac{1}{2}(\dot{v}_{i1}(t) - \dot{v}_{i0}(t)) \hfill & \text{ if } s_{i} \in \mathcal{P} \end{array}\right. $$

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 Time-Step Approximation

The continuous-time 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 \(D_{i}(1-p_{i}^{*})\) 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 i-th 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, ifw₃ = 0,thenh(x + w) − h(x) ∈ [−λ, λ].

Proof

The statement follows from the following expression:

$$h(x + w) - h(x) = w \cdot \vec{n} = \frac{1}{2} (w_{2} - w_{1}) + w_{3} $$

□

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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.

Finally, the reason that individual probability movements are restricted to \({\Delta } = \frac {\lambda }{4}\) is that at the end of the final round, players will move their probabilities and will not be able to respond to subsequent changes in their strategy/payoff states. From the second part of Lemma 6, we can see that in the worst case this can cause a strategy/payoff state to move from the boundary of \(\mathcal {P}^{\lambda /2}\) to the boundary of \(\mathcal {P}^{\frac {3\lambda }{4}} \subset \mathcal {P}^{\lambda }\). However, λ is chosen in such a way so that the worst-case regret within \(\mathcal {P}^{\lambda }\) is at most \(\frac {1}{8}+\alpha \), therefore it follows that UN(α, η) returns a \(\frac {1}{8} + \alpha \) approximate Nash equilibrium. Furthermore, the number of queries is

$$(N + 1) \left( \frac{1024}{\lambda^{3}} \log \left( \frac{8nN}{\eta} \right) \right) = \left( \frac{1}{\lambda} + 1 \right) \left( \frac{1024}{\lambda^{3}} \log \left( \frac{8n}{\lambda \eta} \right) \right). $$

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. □

We return to continuous dynamics to show that we can obtain a worst-case 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 \(\theta \leq \frac {1}{2}\).

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.

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 query-based algorithm that obtains a regret of \(\frac {1}{8} - \frac {1}{220} + \alpha < \frac {1}{8}\) for arbitrary α.

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 well-supported approximate equilibria in cases where c > 1 and for better well-supported 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.