Multi-round cooperative search games with multiple players

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Abstract

A treasure is placed in one of M boxes according to a known distribution and k searchers are searching for it in parallel during T rounds. How can one incentivize selfish players so that the probability that at least one player finds the treasure is maximized? We focus on congestion policies C() specifying the reward a player receives being one of the players that (simultaneously) find the treasure first. We prove that the exclusive policy, in which C(1)=1 and C()=0 for >1, yields a price of anarchy of (1(11/k)k)1, which is the best among all symmetric reward policies. We advocate the use of symmetric equilibria, and show that besides being fair, they are highly robust to crashes of players. Indeed, in many cases, if some small fraction of players crash, symmetric equilibria remain efficient in terms of their group performance while also serving as approximate equilibria.

Introduction

Searching in groups is ubiquitous in multiple contexts, including in the biological world, in human populations as well as on the internet [1], [2], [3]. In many cases there is some prior on the distribution of the searched target. Moreover, when the space is large, each searcher typically needs to inspect multiple possibilities, which in some circumstances can only be done sequentially. This paper introduces a game theoretic perspective to such multi-round treasure hunt searches, generalizing a basic collaborative Bayesian framework previously introduced in [1].

Consider the case that a treasure is placed in one of M boxes according to a known distribution f and that k searchers are searching for it in parallel during T rounds, each specifying a box to visit in each round. Assume w.l.o.g. that the boxes are ordered such that lower index boxes have higher probability to host the treasure, i.e., f(x)f(x+1). We evaluate the group performance by the success probability, that is, the probability that the treasure is found by at least one searcher.

If coordination is allowed, letting searcher i visit box (t1)k+i at time t will maximize success probability. However, as simple as this algorithm is, it is very sensitive to faults of all sorts. For example, if an adversary that knows where the treasure is can crash a searcher before the search starts (i.e., prevent it from searching), then it can reduce the search probability to zero.

The authors of [1] suggested the use of identical non-coordinating algorithms. In such scenarios all processors act independently, using no communication or coordination, executing the same probabilistic algorithm, differing only by the results of their coin flips. As argued in [1], in addition to their economic use of communication, identical non-coordinating algorithms enjoy inherent robustness to different kind of faults. For example, assume that there are k+k searchers, and that an adversary can fail up to k searchers. Letting all searchers run the best non-coordinating algorithm for k searchers guarantees that regardless of which k searchers fail, the overall search efficiency is at least as good as the non-coordinating one for k players. Of course, since k players might fail, any solution can only hope to achieve the best performance of k players. As it applies to the group performance we term this property as group robustness. Among the main results in [1] is identifying a non-coordinating algorithm, denoted A, whose expected running time is minimal among non-coordinating algorithms. Moreover, for every given T, if this algorithm runs for T rounds, it also maximizes the success probability.

The current paper studies the game theoretic version of this multi-round search problem.1 The setting of [1] assumes that the searchers adhere fully to the instructions of a central entity. In contrast, in a game theoretical context, searchers are self-interested and one needs to incentivize them to behave as desired, e.g., by awarding those players that find the treasure first. For many real world contexts, the competitive setting is in fact the more realistic one to assume. Applications range from crowd sourcing [4], multi-agent searching on the internet [5], grant proposals [6], to even contexts of animals [7] (see Appendix A).

In the competitive setting, choosing a good rewarding policy becomes a problem in algorithmic mechanism design [8]. Typically, a reward policy is evaluated by its price of anarchy (PoA), namely, the ratio between the performances of the best collaborative algorithm and the worst equilibrium [9]. Aiming to both accelerate the convergence process to an equilibrium and obtain a preferable one, the announcement of the reward policy can be accompanied by a proposition for players to play particular strategies that form a profile at equilibrium.

This paper highlights the benefits of suggesting (non-coordinating) symmetric equilibria in such scenarios, that is, to suggest the same non-coordinating strategy to be used by all players, such that the resulting profile is at equilibrium. This is of course relevant assuming that the price of symmetric stability (PoSS), namely, the ratio between the performances of the best collaborative algorithm and the best symmetric equilibrium, is low. Besides the obvious reasons of fairness and simplicity, from the perspective of a central entity who is interested in the overall success probability, we obtain the group robustness property mentioned above, by suggesting that the k+k players play according to the strategy that is a symmetric equilibrium for k players. Obviously, this group robustness is valid only provided that the players indeed play according to the suggested strategy. However, the suggested strategy is guaranteed to be an equilibrium only for k players, while in fact, the adversary may keep some of the extra k players alive. Interestingly, however, in many cases, a symmetric equilibrium for k players also serves as an approximate equilibrium for k+k players, as long as kk. As we show, this equilibrium robustness property is rather general, holding for a class of games, that we call monotonously scalable games.

A treasure is placed in one of M boxes according to a known distribution f and k players are searching for it in parallel during T rounds. Assume w.l.o.g. that f(x)>0 for every x and that f(x)f(x+1).

An execution of T rounds is a sequence of box visitations σ=x(1),x(2),x(T), one for each round iT. We assume that a player visiting a box has no information on whether other players have already visited that box or are currently visiting it. Hence, a strategy of a player is a probability distribution over the space of executions of T rounds. Note that the probability of visiting a box x in a certain round may depend on the boxes visited by the player until this round, but not on the actions of other players. A strategy is non-redundant if at any given round it always checks a box it didn't check before (as long as there are such boxes).

A profile is a collection of k strategies, one for each player. Special attention will be devoted to symmetric profiles. In such profiles all players play the same strategy (note that their actual executions may be different, due to different probabilistic choices).

While slightly abusing notation, we shall associate each strategy A with its probability matrix,A:{1,,M}×{1,,T}[0,1], where A(x,t) is the probability that strategy A visits x for the first time at round t. We also denote A˜(x,t) as the probability that A does not visit x by, and including, time t. That is, A˜(x,t)=1stA(x,t) and A˜(x,0)=1. For convenience we denote by δx,t the matrix of all zeros except 1 at x,t. Its dimensions will be clear from context.

A profile is evaluated by its success probability, i.e., the probability that at least one player finds the treasure by time T. Formally, let P be a profile. Then,success(P)=xf(x)(1APA˜(x,T)). The expected running time in the symmetric case, which is xf(x)tA˜(x,t)k, was studied in [1]. That paper identified a strategy, denoted A, that minimizes this quantity. In fact, it does so by minimizing the term xf(x)A˜(x,t)k for each t separately. Note that minimizing the case t=T is exactly the same as maximizing the success probability. Thus, restricted to the case where all searchers use the same strategy, A simultaneously optimizes the success probability as well as optimizes the expected running time. A description of A is provided below.

We note that in [1] the matrix of A is given, and then an algorithm is explicitly described that has its matrix (Section 4.3 in [1]). We describe the matrix only, as its details are necessary for this paper. Denote q(x)=f(x)1/(k1). For each t,A˜(x,t)=min(1,α(t)q(x)), where α(t)0 is such that xA˜(x,t)=Mt. Of course, once A˜ is known, then so is A: A(x,t)=A˜(x,t1)A˜(x,t).

A natural way to incentivize players is by rewarding those players that find the treasure before others. A congestion policy C() is a function specifying the reward that a player receives if it is one of players that (simultaneously) find the treasure for the first time. We assume that C(1)=1, and that C is non-negative and non-increasing. Due to the fact that the policy in which C1, i.e., C()=1 for every , is rather degenerate, we henceforth assume that there exists an such that C()<1. We shall give special attention to the following policies.

  • The sharing policy is defined by Cshare()=1/, namely, the treasure is shared equally among all those who find it first.

  • The exclusive policy is defined by Cex(1)=1, and Cex()=0 for >1, namely, the treasure is given to the first one that finds it exclusively; if more than one discover it, they get nothing.2

Note that we allow for C() to be greater than 1. For example, a competition may give 1$ to all winners, as long as there are at most 3. If it would give 1$ no matter how many winners there are, we would be in the policy C1.

A configuration is a triplet (C,f,T), where C is a congestion policy, T is a positive integer, and f is a positive non-increasing probability distribution on M boxes.

Let (C,f,T) be a configuration. The value of box x at round t when playing against3 a profile P is the expected gain from visiting x at round t. Formally,vP(x,t)=f(x)=0k1C(+1)Pr(xwas not visited before timet, andat timetis visited by players of P)=f(x)=0k1C(+1)|I|=IPAIA(x,t)AIA˜(x,t). We can then define utility of A in round t and the utility of A as:UP(A,t)=xA(x,t)vP(x,t),UP(A)=tUP(A,t). Here are some specific cases we are interested in:

  • For symmetric profiles, vA(x,t) denotes the value when playing against k1 players playing A. In this case,vA(x,t)=f(x)=0k1C(+1)(k1)A(x,t)A˜(x,t)k1. UA(B,t) and UA(B) are defined in an analogous way in this case.

  • For the exclusive policy,vP(x,t)=f(x)APA˜(x,t).

  • For the exclusive policy in symmetric profiles,vA(x,t)=f(x)A˜(x,t)k1.

Denote PA is the set of players of P excluding A. A profile P is a Nash equilibrium under a configuration, if for any AP and any other strategy B, UPA(A)UPA(B). Similarly, a strategy A is called a symmetric equilibrium if the profile Ak consisting of all k players playing according to A is an equilibrium. We also define approximate equilibria:

Definition 1

For ϵ>0, we say a profile P is a (1+ϵ)-equilibrium if for every AP and for every strategy B, UPA(B)(1+ϵ)UPA(A).

Both the expressions for the success probability and utility solely depend on the values of the probability matrices associated with the strategies in question. Hence we view all strategies sharing the same matrix as equivalent. Note that a matrix does not necessarily correspond to a unique strategy, as illustrated by the following equivalent strategies, for which A(x,t)=B(x,t)=1/M for every tM and 0 thereafter:

  • Strategy A chooses uniformly at every round one of the boxes it didn't choose yet.

  • In the first round, strategy B chooses a random x{0,,M1}. Then, at each round t1 it visits box (x+tmodM)+1.

Matrices are much simpler to handle than strategies, and so we would rather think of our game as a game of probability matrices than a game of strategies. For this we need to characterize which matrices are indeed probability matrices of strategies. Clearly, a probability matrix is non-negative. Also, by their definition, each row and each column sums to at most 1. Such a matrix is called doubly-substochastic. It turns out that these conditions are sufficient. That is, every doubly-substochastic matrix is a probability matrix of some strategy.

A doubly-substochastic matrix is a partial permutation if it consists of only 0 and 1 values. The following is a generalization of the Birkhoff - von Neumann theorem, proved for example in [11].

Theorem 2

A matrix is doubly-substochastic iff it is a convex combination of partial permutations.

Furthermore, Birkhoff's construction [12] finds this decomposition in polynomial time, and guarantees it contains at most a number of terms as the number of positive elements of the matrix. The generalization of [11] does not change this claim significantly, as it embeds the doubly-substochastic matrix in a doubly-stochastic one which is at most 4 times larger.

Corollary 3

If matrix A is doubly-substochastic then there is some strategy such that A is its probability matrix. Furthermore, this strategy can be found in polynomial time, and is implementable as a polynomial algorithm.

Proof

First note that the claim is true if A is a partial permutation. The strategy in this case will be a deterministic strategy, which may sometimes choose not to visit any box. In the general case, Theorem 2 states that there exist θ1,θ2,,θk, such that i=1kθi=1 and partial permutations A1,A2,,Ak, such that A=i=1kθiAi. As mentioned, each Ai is the probability matrix of some strategy Bi. Define the following strategy B as follows: with probability θi run strategy Bi. Then, the probability matrix of B is iθiAi=A, as required. 

We will therefore view our game as a game of doubly-substochastic matrices.

Informally, a strategy is greedy at a round if its utility in this round is the maximum possible in this round. Formally, given a profile P and some strategy A, we say that A is greedy w.r.t. P at time t if for any strategy B such that for every x and s<t, B(x,s)=A(x,s), we have UP(A,t)UP(B,t). We say A is greedy w.r.t. P if it is greedy w.r.t. P for each tT. A strategy A is called self-greedy (or sgreedy for short) if it is greedy w.r.t. the profile with k1 players playing A.

Let (C,f,T) be a configuration. Denote by Nash(C,f,T) the set of equilibria for this configuration, and by S-Nash(C,f,T) the subset of symmetric ones. Let P(T) be the set of all profiles of T-round strategies. We are interested in the following measures.

  • The Price of Anarchy (PoA) isPoA(C,f,T):=maxPP(T)success(P)minPNash(C,f,T)success(P).

  • The Price of Symmetric Stability (PoSS) isPoSS(C,f,T):=maxPP(T)success(P)maxAS-Nash(C,f,T)success(Ak).

  • The Price of Symmetric Anarchy (PoSA) isPoSA(C,f,T)=maxPP(T)success(P)minAS-Nash(C,f,T)success(Ak).

The setting of multi-rounds poses several challenges that do not exist in the single round game. An important one is the fact that, in contrast to the single round game, the multi-round game is not a potential game. Indeed, being a potential game has several implications, and a significant one is that such a game always has a pure equilibrium. However, we show that multi-round games do not always have pure equilibria and hence are not potential games.

Claim 4

The single-round game is a potential game, yet the multi-round game is not.

Proof

For the single round, assume that P is a deterministic profile, and let xP be the number of players that choose box x in P. Denote,Φ(P)=xf(x)=1xPC(), where we use the convention that =10C()=0. If a player changes strategy and chooses (deterministically) some box y instead of box x, then the change in its utility is f(y)C(yP+1)f(x)C(xP). This is also the change that Φ(P) sees. This extends naturally to mixed strategies, and so the single round game is a potential game.

On the other hand, the multi-round game does not always have a pure equilibrium, and so is not a potential game. For example, the following holds for any policy C. Consider the case of a three boxes (M=3), two rounds (T=2), two players (k=2), and all boxes have an equal probability of holding the treasure (f(x)=1/3). Note that C(2)<1 since here k=2 and we assumed C1.

Assume there is some deterministic profile that is at equilibrium, and w.l.o.g. assume player 1's first pick is box 1. There are two cases:

  • 1.

    Player 1 picks it again in the second round. Player 2's strictly best response is to pick box 2 and then 3 (or the other way around). In this case, player 1 would earn more by first picking box 3 (box 2) and then box 1. In contradiction.

  • 2.

    Player 1 picks a different box in the second round. W.l.o.g. assume it is box 2. Player 2's strictly best response is to first take box 2 and then take box 3. However, player 1 would then prefer to start with box 3 and then box 1. Again a contradiction.  

Another important difference is that for policies that incur high levels of competition (such as the exclusive policy), profiles that maximize the success probability are at equilibrium in the single round case, whereas they are not in the multi-round game.

In the single-round game, when Mk, the success probability is maximized when each player exclusively visits one box in 1,2,,k with probability 1. Under the exclusive policy, for example, such a profile is also at equilibrium. In fact, if f(k)f(1)/2 then the same is true also for the sharing policy.

For the multi-round setting, when MTk, an optimal scenario is also achieved by a deterministic profile, e.g., when player i visits box i+(t1)k in round t. However, this profile would typically not be an equilibrium, even under the exclusive policy. Indeed, when f(x) is strictly decreasing, player 2 for example, can gain more by stealing box k+1 from player 1 in the first round, then safely taking box 2 in the second round, and continuing from there as scheduled originally. This shows that in the multi-round game, the best equilibrium has only sub-optimal success probability.

We first provide a simple, yet general, robustness result, that holds for symmetric (approximate) equilibria in a family of games, termed monotonously scalable. Informally, these are games in which the sum of utilities of players can only increase when more players are added, yet for each player, its individual utility can only decrease. Our search game with the sharing policy is one such example.

Theorem 5

Consider a symmetric monotonously scalable game. If A is a symmetric (1+ϵ)-equilibrium for k players, then it is an (1+ϵ)(1+/k)-equilibrium when played by k+ players.

Theorem 5 is applicable in fault tolerant contexts. Consider a monotonously scalable game with k+k players out of which at most k may fail. Let Ak be a symmetric (approximate) equilibrium designed for k players and assume that its social utility is high compared to the optimal profile with k players. The theorem implies that if players play Ak, then regardless of which k players fail (or decline to participate), the incentive to switch strategy would be very small, as long as kk. Moreover, due to symmetry, if the social utility of the game is monotone, then the social utility of Ak when played with k players is guaranteed when playing with more. Thus, in such cases we obtain both group robustness and equilibrium robustness.

Coming back to our search game, we consider general policies, focus on symmetric profiles, and specifically, on the properties of sgreedy strategies.

Theorem 6

For every policy C there exists a non-redundant sgreedy strategy. Moreover, all such strategies are equivalent and are symmetric (1+C(k))-equilibria.

When C(k)=0 this shows that a non-redundant sgreedy strategy is actually a symmetric equilibrium. We next claim that this is the only symmetric equilibrium (up to equivalence).

Claim 7

For any policy such that C(k)=0, all symmetric equilibria are equivalent.

Theorem 6 is non-constructive because it requires calculating the inverse of non-trivial functions. Therefore, we resort to an approximate solution.

Theorem 8

Given θ>0, there exists an algorithm that takes as input a configuration, and produces a symmetric (1+C(k))(1+θ)-equilibrium. The algorithm runs in polynomial time in T, k, M, log(1/θ), log(1/(1C(k))), and log(1/f(M)).

Recall that the exclusive policy is defined by Cex(1)=1 and Cex()=0 for every >1. Recall also that A is the symmetric strategy that, as established in [1], gives the optimal success probability among symmetric strategies (see Section 1.1.4). We show that A is a non-redundant and sgreedy strategy w.r.t. the exclusive policy. Hence, Theorem 6 implies the following.

Theorem 9

Under the exclusive policy, Strategy A of [1] is a symmetric equilibrium.

Claim 7 together with the fact that A has the highest success probability among symmetric profiles, implies that both the PoSS and the PoSA of Cex are optimal (and equal) on any f and T when compared to any other policy. The next theorem considers general equilibria.

Theorem 10

Consider the exclusive policy. For any profile Pnash at equilibrium and any symmetric profile A, success(Pnash)success(A).

Observe that, as A is a symmetric equilibrium, Theorem 10 provides an alternative proof for the optimality of A (established in [1]). Interestingly, this alternative proof is based on game theoretic considerations, which is a very rare approach in optimality proofs.

Combining Theorem 9, Theorem 10, we obtain:

Corollary 11

For any f and T, PoA(Cex,f,T)=PoSA(Cex,f,T). Moreover, for any policy C, PoA(Cex,f,T)PoA(C,f,T).

At first glance the effectiveness of Cex might not seem so surprising. Indeed, it seems natural that high levels of competition would incentivize players to disperse. However, it is important to note that Cex is not extreme in this sense, as one may allow congestion policies to also have negative values upon collisions. Moreover, one could potentially define more complex kinds of policies, e.g., policies that depend on time, and reward early finds more. However, the fact that A is optimal among all symmetric profiles combined with the fact that any symmetric policy has a symmetric equilibrium [13] implies that no symmetric reward mechanism can improve either the PoSS, the PoSA, or the PoA of the exclusive policy.

We proceed to show a tight upper bound on the PoA of Cex. Note that as k goes to infinity the bound converges to e/(e1)1.582.

Theorem 12

For every T, supfPoA(Cex,f,T)=(1(11/k)k)1.

Concluding the results on the exclusive policy, we study the robustness of A. Let Ak denote algorithm A when set to work for k players. Unfortunately, for any ϵ, there are cases where Ak is not a (1+ϵ)-equilibrium even when played by k+1 players. However, as indicated below, A is robust to failures under reasonable assumptions regarding the distribution f.

Theorem 13

If f(1)f(M)(1+ϵ)k1k, then Ak is a (1+ϵ)-equilibrium when played by k+k players.

Another important policy to consider is the sharing policy. This policy naturally arises in some circumstances, and may be considered as a less harsh alternative to the exclusive one. Although not optimal, it follows from Vetta [14] that its PoA is at most 2 (see Appendix C). Furthermore, as this policy yields a monotonously scalable game, a symmetric equilibrium under it is also robust. Therefore, the existence of a symmetric profile which is both robust and has a reasonable success probability is guaranteed.

Unfortunately, we did not manage to find a polynomial algorithm that generates a symmetric equilibrium for this policy. However, Theorem 8 gives a symmetric (1+θ)(1+1/k)-equilibrium in polynomial time for any θ>0. This strategy is also robust thanks to Theorem 5. Moreover, the proof in [14] regarding the PoA can be extended to hold for approximate equilibria. In particular, if P is some (1+ϵ)-equilibrium w.r.t. the sharing policy, then for every f and T, success(P)12+ϵmaxPP(T)success(P). The proof of this claim appears in Appendix C).

Fault tolerance has been a major topic in distributed computing for several decades, and in recent years more attention has been given to these concepts in game theory [15], [16]. For example, Gradwohl and Reingold studied conditions under which games are robust to faults, showing that equilibria in anonymous games are fault tolerant if they are “mixed enough” [17].

Restricted to a single round the search problem becomes a coverage problem, which has been investigated in several papers. For example, Collet and Korman studied in [18] (one-round) coverage while restricting attention to symmetric profiles only. The main result therein is that the exclusive policy yields the best coverage among symmetric profiles. Gairing [19] also considered the single round setting, but studied the optimal PoA of a more general family of games called covering games (see also [10], [20]). Motivated by policies for research grants, Kleinberg and Oren [6] considered a one-round model similar to that in [18]. Their focus however was on pure strategies only. The aforementioned papers give a good understanding of coverage games in the single round setting. As mentioned, however, the multi-round setting studied here is substantially more complex than the single-round setting.

The multi-armed bandit problem [21], [22] is commonly recognized as a central exemplar of the exploration versus exploitation trade-off. In the classic version, an agent (gambler) chooses sequentially between alternative arms (machines), each of which generates rewards according to an unknown distribution. The objective of the agent is to maximize the sum of rewards earned during the execution. In each round, the dilemma is between “exploitation”, i.e., choosing the arm with the highest expected payoff, and “exploration”, i.e., choosing an arm in order to acquiring knowledge that can be used to make better-informed decisions in the future. Most of the literature concerning the multi-armed bandit problem considers a single agent. This drastically differs from our setting that concerns multiple competing players. Furthermore, in our setting, information about the quality of boxes (arms) in known in advance, and there is no use in returning to a box (arm) after visiting it.

The area of “incentivized bandit exploration”, introduced by Kremer et al. [23], studies incenticizing schemes in the context of the tradeoff between exploration and exploitation [24], [25], [26]. In this version of multi-armed bandits problem, the action decisions are controlled by multiple self-interested agents, based on recommendations given by a centralized algorithm. Typically, in each round, an agent arrives, chooses an arm to pull among several alternatives, receives a reward, and leaves forever. Each agent “lives” for a single round and therefore pulls the arm with the highest current estimated payoff, given its knowledge. The centralized algorithm controls the flow of information, which in turn can incentivize the agents to explore. This is different from in our scenario in which agents remain throughout rounds, and know their entire history. Moreover, in our setting, the central entity controls only the congestion policy, and does not manipulate the information agents have throughout the execution. Within the scope of incentivized bandit exploration, related models have been studied in [27], [28] while considering time-discounted utilities, and in [29] studying the notion of fairness. A extensive review of this line of research can be found in Chapter 11 of Slivkins [22].

The settings of selfish routing, job scheduling, and congestion games [30], [31] all bear similarities to the search game studied here, however, the social welfare measurements of success probability or running time are very different from the measures studied in these frameworks, such as makespan or latency [32], [33], [8], [34].

Section snippets

Robustness in symmetric monotonously scalable games

Consider a symmetric game where the number of players is not fixed. Let UP(A) denote the utility that a player playing A gets if the other players play according to P and let σ(P)=APUPA(A). We say that such a game is monotonously scalable if:

  • 1.

    Adding more players can only increase the sum of utilities, i.e., if PP then σ(P)σ(P).

  • 2.

    Adding more players can only decrease the individual utilities, i.e., if PP then for all AP, UPA(A)UPA(A).

Theorem 5

Consider a symmetric monotonously scalable game. If

General policies

A first useful observation is that if a box has some probability of not being chosen, then it has a positive value. This is clear from the definition of utility, from the fact that C(1)=1 and because C is non-negative.

Observation 14

If for all AP, A˜(x,t)>0, then vP(x,t)>0.

The exclusive policy

Lemma 35

Under the exclusive policy, A restricted to TM rounds is sgreedy and non-redundant.

Proof

See Section 1.1 for a description of A. To see it is non-redundant:xA(x,t)=xA˜(x,t1)xA˜(x,t)=(Mt1)(Mt)=1. To see that it is sgreedy, let us fix t. The value of a box x isvA(x,t)=f(x)A˜(x,t)k1=f(x)min(1,α(t)q(x))k1=min(f(x),α(t)k1). If A(x,t)>0 then A˜(x,t)<1, which means that vA(x,t)=α(t)k1, and so is constant for all such x. If A(x,t)=0 then A˜(x,t)=1, and so α(t)1/q(x)=f(x)1/(k1).

Future work and open questions

In [1], the main complexity measure was actually the running time and not the success probability. Our results about equilibria are also relevant to this measure, but the social gain is different. For example, it is still true that A is an equilibrium under the exclusive policy, and that all other symmetric equilibria in the exclusive policy are equivalent to it. As A is optimal among symmetric profiles w.r.t. the running time, the PoSA of Cex is equal to the PoSS, and it is also the best

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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    This work has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 648032).

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