Agent Failures in Totally Balanced Games and Convex Games

Transcription

1 Agent Failures in Totally Balanced Games and Convex Games Yoram Bachrach 1, Ian Kash 1, and Nisarg Shah 2 1 Microsoft Research Ltd, Cambridge, UK. {yobach,iankash}@microsoft.com 2 Comuter Science Deartment, Carnegie Mellon University, USA. nkshah@cs.cmu.edu Abstract. We examine the imact of indeendent agents failures on the solutions of cooerative games, focusing on totally balanced games and the more secific subclass of convex games. We follow the reliability extension model, recently roosed in [1] and show that a (aroximately) totally balanced (or convex) game remains (aroximately) totally balanced (or convex) when indeendent agent failures are introduced or when the failure robabilities increase. One imlication of these results is that any reliability extension of a totally balanced game has a nonemty core. We roose an algorithm to comute such a core imutation with high robability. We conclude by outlining the effect of failures on non-emtiness of the core in cooerative games, esecially in totally balanced games and simle games, thereby extending observations in [1]. Keywords: Totally Balanced Games, Convex Games, Agent Failures, Cooerative Game Theory 1 Introduction Consider a communication network designed to transmit information from a source node to a target node, where selfish agents control the different links in the network. Suose the utility generated by the network is roortional to the bandwidth it can achieve between the source and the target. Further suose that the links are not fully reliable and may fail, and that these link failures are indeendent of each other, although the failure robability of each link may be different. The surviving links rovide a certain bandwidth from the source to the target. Since it is not known a riori which links would fail, there is uncertainty regarding the revenue that the agents would generate. In such a setting the agents owning the links tyically need each other in order to generate revenue, but since they are selfish each of them attemts to maximize his own share of the revenue. Which agreements are these agents likely to make regarding sharing the revenue? How do the link failures affect the agents ability to reach a stable agreement regarding distributing the gains amongst themselves? Can we comute such stable ayment allocations?

2 2 Yoram Bachrach, Ian Kash, and Nisarg Shah Interactions between selfish agents who must cooerate to achieve their goals are analyzed using cooerative game theory [18,7], where solution concets attemt to characterize how such agents might agree to act and share the resulting gains among themselves. A rominent such concet is the core [11] which requires that no sub-coalition could defect and imrove its utility by oerating on its own. Shaley and Shubik introduced the class of totally balanced games and showed that such games have a non-emty core [24]. Many interesting and ractical classes of games have been shown to be totally balanced, such as the network flow game [13], Owen s linear roduction game [19], the ermutation game [27], the assignment game [25], the minimum cost sanning tree game [12], etc. Our motivating examle is a network flow game with indeendent agent failures. Desite the wide coverage of cooerative interactions, most models ignore failures although it is hardly realistic to assume that all agents can always fill their roles. We use the reliability extension model [1] which formalizes indeendent agent failures in cooerative games, to investigate such failures in totally balanced games and in the more secific subclass of convex games. Our Contribution: We first study the reliability extension of general cooerative games. We show how a game is transformed when failures are introduced or when the reliabilities (robabilities of agents not failing) change. Next we introduce the class of ɛ-totally balanced games, a natural generalization of totally balanced games, and investigate the effect of failures on such games. We rove that every reliability extension of an ɛ-totally balanced game is also ɛ-totally balanced. For ɛ = 0, this imlies that every reliability extension of a totally balanced game is also totally balanced. Further, we show that decreasing one or more reliabilities in an ɛ-totally balanced game kees it ɛ-totally balanced. Using Shaley s result that convex games are totally balanced [23], our results imly that every reliability extension of a convex game is also totally balanced. This strengthens a result by Bachrach et. al. [1] who rove that every reliability extension of a convex game has a non-emty core. Similarly to ɛ-totally balanced games, we also introduce ɛ-convex games and rove that every reliability extension of an ɛ-convex game is also ɛ-convex. For ɛ = 0, this imlies that every reliability extension of a convex game is not just totally balanced, but convex. Further, we show that decreasing reliabilities in an ɛ-convex game kees it ɛ- convex. We then rove that any ɛ/(n 1)-convex game is ɛ-totally balanced, generalizing Shaley s result that convex games are totally balanced. Additionally, we examine the comutational asects of a game s reliability extension, and rovide an algorithm that comutes a core solution of any reliability extension of a totally balanced game with high robability. Our results show that both introducing failures and increasing failure robabilities reserve core non-emtiness in totally balanced games. We oint out that neither of these reserve core non-emtiness in general cooerative games. Bachrach et. al. [1] observe that introducing failures reserves non-emtiness of the core in simle games (where every coalition has value either 0 or 1). Surrisingly, we show that this is not the case for increasing failure robabilities.

3 2 Related Work Agent Failures in Totally Balanced Games and Convex Games 3 Shaley and Shubik [24] introduced the notion of totally balanced games and roved their equivalence to the class of market games. Kalai and Zemel [13] later roved that they are also equivalent to two other classes of games: finite collections of simle additive games and network flow games. Owen [19] and Tijs et. al. [27] introduced two ractical classes of totally balanced games: resectively, linear roduction games arising from linear rogramming roblems and ermutation games arising in sequencing and assignment roblems. Deng et. al. [10] extended the analysis of total balancedness to various combinatorial otimization games, artition games and acking and convering games. These results suggest that the class of totally balanced games is elementary and ractical. Our analysis follows the reliability extension model of Bachrach et. al. [1] to examine the imact of indeendent agent failures in totally balanced games. A somewhat reminiscent model was roosed by Chalkiadakis et. al. [8] in which they consider the roblem of coalition formation in a Bayesian setting. In their model, agents have tyes which are rivate information and agents have beliefs about the tyes of the other agents. In our setting, the failure robabilities can be viewed as tyes, but we focus on the secific case when these failure robabilities are ublic information and the failures are indeendent. Agent failures have also been widely studied in non-cooerative game theory. For examle, Penn et. al. [20] study indeendent agent failures in congestion games, which are noncooerative normal form games. Such failures have also been studied in other fields such as reliable network formation [4], non-cooerative Nash networks [5], sensor networks [14] etc. It is quite surrising that such an elementary notion of failure was only recently formalized in cooerative games. 3 Preliminaries A transferable utility cooerative game G = (N, v) is comosed of a set of agents N = {1, 2,..., n} and a characteristic function v : 2 N R indicating the total utilities achievable by various coalitions (subsets of agents). By convention, v( ) = 0. For any agent i N and coalition S N, we denote S {i} by S + i and S \ {i} by S i. For a game G = (N, v) and coalition S N, G S denotes the subgame of G obtained by restricting the set of agents to S. Convex Games: A characteristic function is called suermodular if for each i N and for all S and T such that S T N i, we have v(s + i) v(s) v(t + i) v(t ) (i.e., increasing marginal returns). A game is called convex if its characteristic function is suermodular. Similarly, for any ɛ 0, a characteristic function is called ɛ-suermodular if for each i N and for all S and T such that S T N i, we have v(s + i) v(s) v(t + i) v(t ) + ɛ. Define a game to be ɛ-convex if its characteristic function is ɛ-suermodular. Note that convex games are recovered as the secial case of ɛ = 0.

4 4 Yoram Bachrach, Ian Kash, and Nisarg Shah Imutation: The characteristic function defines the value that a coalition can achieve on its own, but not how it should distribute the value among its members. A ayment vector = ( 1,..., n ) is called a re-imutation if n i=1 i = v(n). A ayment vector = ( 1,..., n ) is called an imutation if it is a reimutation and also individually rational, i.e., i v({i}) for every i N. Here, i is the ayoff of agent i, and the ayoff of a coalition C is (C) = i C i. Core: A basic requirement for any good imutation is that the ayoff to every coalition is at least as much it can gain on its own so that no coalition can gain by defecting. The core is the set of all imutations such that (N) = v(n) and (S) v(s) for all S N. It may be emty or may contain more than one imutation. One closely related concet is that of ɛ-core. For any ɛ R, the ɛ-core is the set of all imutations such that (N) = v(n) and for every S N such that S N, we have (S) v(s) ɛ. When ɛ > 0, it serves as a relaxation of the core and is useful in redicting behaviour in games where the core is emty. When ɛ < 0, it serves as a stronger concet where every coalition requires at least an incentive of ɛ to defect. We denote the case of ɛ > 0 as the aroximate core and the case of ɛ < 0 as the suerstable core. For any game, it is easy to show that the set {ɛ the ɛ-core is non-emty} is comact and thus has a minimum element ɛ min. The ɛ min is known as the least core value of the game and the ɛ min -core is known as the least core. Total Balancedness: As defined by Shaley and Shubik [24], a game is called totally balanced if every subgame of the game has a non-emty core. We define a natural generalization of totally balanced games. For any ɛ 0, a game is called ɛ-totally balanced if every subgame of the game has non-emty ɛ-core. Reliability Game: As defined in [1], a reliability game G = (N, v, r) consists of the set of agents N = {1, 2,..., n}, the base characteristic function v which describes the values of the coalitions in the absence of failures, and the reliability vector r where r i is the robability of agent i not failing. After taking failures into account, the characteristic function of the reliability game, denoted by v r, is given by the following equation. For every coalition S N, v r (S) = Pr(S S) v(s ) = r i (1 r j ) v(s ). (1) S S S S i S j S\S Here, Pr(S S) denotes the robability that every agent in S survives and every agent in S \S fails so v r (S) is the exected utility S achieves under failures. The set S is called the survivor set for the coalition S. For the base game G = (N, v), the game G r = (N, v, r) is called the reliability extension of G with the reliability vector r. For a reliability vector r, we denote by r -i the vector of reliabilities of all agents excet i and by r = (, r -i ) the reliability vector where r i = and r j = r j for j i. For vectors x and y, define x y if x i y i for every i.

5 Agent Failures in Totally Balanced Games and Convex Games 5 4 Reliabilities, Total Balancedness and Convexity We examine how the value of a coalition changes as the reliability of an agent changes in a general game. Lemma 1. Let G = (N, v, r) be a reliability game. Let i N be an agent and let = r i > 0 be the reliability of agent i in G. Take 0 1 and let G = (N, v, r ) where r = (, r -i ). Let v r and v r be the characteristic functions of G and G resectively. Then the following holds For any coalition S N such that i / S, we have v r (S) = v r (S). 2. For any coalition S N such that i S, we have ) v r (S) = vr (S) + (1 v r (S i). Proof Sketch. Part 1 of the roof is trivial and follows directly from Equation (1). For the second art, for any coalition S N such that i S, we define vi rel (S) as the value of S in the game G rel i = (N, v, ri rel ) where ri rel = (1, r -i ). Now we break the summation in Equation (1) into two arts: summation over the subsets containing i and summation over the subsets not containing i, and observe that v r (S) = vi rel (S) + (1 ) v r (S i) and similarly, v r (S) = vi rel (S) + (1 ) v r (S i). Finally, observing that v r (S i) = v r (S i) (using art 1) and eliminating vi rel (S) from the two equations, we get the desired result. The roof aears in the full version of the aer. 4 Note that by starting with G = (N, v, 1) where 1 = 1, 1,..., 1, we can use Lemma 1 to analyze the effect of introducing failures into a cooerative game as well. 4.1 Aroximately Totally Balanced Games We now analyze the reliability extension of ɛ-totally balanced games and rove that ɛ-total balancedness is reserved when the reliability of one agent decreases. Theorem 1. Let ɛ 0 and G = (N, v, r) be a reliability game that is ɛ-totally balanced. Fix i N and let = r i > 0 be the reliability of agent i in G. Take such that 0 and define G = (N, v, r ) where r = (, r -i ). Then G is ɛ-totally balanced. Proof. Let v r and v r denote the characteristic functions of G and G resectively. We want to rove that G is ɛ-totally balanced, i.e., for any coalition 3 The equation in art 2 of Lemma 1 really catures both arts 1 and 2. Part 1 is obtained by observing that S i = S when i / S. The two cases are searated for clarity and for the convenience of the reader. 4 The full version is available from: htt://

6 6 Yoram Bachrach, Ian Kash, and Nisarg Shah S N, the subgame G S has an ɛ-core imutation. Fix any coalition S N. There are two cases: i S and i / S. 5 If i / S, then using Lemma 1 we see that v r (C) = v r (C) for every C S, i.e., the subgames G S and G S are equivalent. Further, since G is an ɛ-totally balanced game, its subgame G S has an ɛ-core imutation x. It is easy to see that x is also an ɛ-core imutation of G S. Now let i S. Since G is an ɛ-totally balanced game, both its subgames G S and G S i have ɛ-core imutations, say x S and x S i (take x φ = 0, 0,..., 0 ). Extend both vectors by setting the ayments to agents in N \ S (and ayment to i in x S i ) to be zero. We rove that x = / x S + (1 /) x S i is an ɛ-core imutation of G S. First we show that x is a re-imutation. ] x(s) = x S(S) + [1 = vr (S) + [1 x S i (S) = x S(S) + ] v r (S i) = v r (S), ] [1 x S i (S i) Here, the second transition follows since the ayment to agent i in x S i is 0, the third transition follows since x S and x S i are ɛ-core imutations of G S and G S i resectively and the last transition follows from Lemma 1. Now for any coalition C S and C S, we want to show that x(c) v r (C) ɛ. We again take two cases: i C and i / C. Let i / C, i.e., C S i. Since x S and x S i are ɛ-core imutations of G S and G S i resectively, we have that x S (C) v r (C) ɛ and x S i (C) v r (C) ɛ. Therefore, ] [ ] x(c) = x S(C)+ [1 x S i (C) + 1 (v r (C) ɛ) = v r (C) ɛ, where the second transition uses and the third transition follows since v r (C) = v r (C) (art 1 of Lemma 1). Now let i C. Once again, we have that x(c) = x S(C) + [1 (vr (C) ɛ) + ] x S i (C) = [1 x S(C) + ] (v r (C i) ɛ) = v r (C) ɛ. ] [1 x S i (C i) The second transition follows since ayment to agent i in x S i is 0. The third transition follows since x S and x S i are ɛ-core imutations of G S and G S i resectively and since. The last transition follows due to art 2 of Lemma 1. Hence, we roved that x(s) = v r (S) and x(c) v r (C) ɛ for every coalition C S where C S. This roves that x is an ɛ-core imutation of G S. Since S N was selected arbitrarily, we have roved that every subgame of G has non-emty ɛ-core, i.e., G is ɛ-totally balanced. 5 It is ossible to combine all the cases in the roof of Theorem 1 using Footnote 3. However, a case-wise analysis is resented to avoid any confusion.

7 Agent Failures in Totally Balanced Games and Convex Games 7 Since ɛ-total balancedness is reserved when we decrease a single reliability, we can decrease multile reliabilities one-by-one and reeatedly aly Theorem 1 to show ɛ-total balancedness in the resulting game, so we obtain the following. Corollary 1. Let ɛ 0. Let G = (N, v, r) be an ɛ-totally balanced reliability game and G = (N, v, r ) where r r. Then G is ɛ-totally balanced. Any reliability extension of a game can be obtained by starting from the base game (equivalent to its reliability extension with the reliability vector 1) and decreasing reliabilities as required. Hence Corollary 1 imlies that reliability extensions reserve ɛ-total balancedness. Conversely if a game G is not ɛ-totally balanced, it has a subgame G S, which is also a reliability extension with reliability 1 for i S and 0 otherwise, having emty ɛ-core. This roves the following. Corollary 2. For any ɛ 0, a game is ɛ-totally balanced if and only if every reliability extension of the game is ɛ-totally balanced. Shaley [23] showed that convex games are totally balanced. Thus Corollary 2 imlies that every reliability extension of a convex game is totally balanced. This strengthens a theorem by Bachrach et. al. [1] which states that every reliability extension of a convex game has a non-emty core. 6 We strengthen this further and rove that every reliability extension of a convex game is in fact convex. 4.2 Aroximately Convex Games For the reliability extension of ɛ-convex games, the results are arallel to those for ɛ-totally balanced games, but require different roof techniques. Theorem 2. Let ɛ 0. Let G = (N, v, r) be an ɛ-convex reliability game. Fix i N and let = r i > 0 be the reliability of agent i in G. Take such that 0 and define G = (N, v, r ) where r = (, r -i ). Then G is ɛ-convex. Proof Sketch. Let v r and v r be the characteristic functions of G and G resectively. We want to rove that G is ɛ-convex, i.e., for every j N and for all S T N j, v r (S + j) v r (S) v r (T + j) v r (T ) + ɛ. We know that this is true for v r since G is ɛ-convex. We analyze the marginal contributions of j to S and T in both v r and v r and aly ɛ-convexity of G and Lemma 1 (wherever required) in order to rove ɛ-convexity of G. The roof aears in the full version of the aer. Similarly to totally balanced games, Theorem 2 can be extended to cover general decreases in reliabilities, including those starting from the base game. Corollary 3. Let ɛ 0. Let G = (N, v, r) be an ɛ-convex reliability game and G = (N, v, r ) where r r. Then G is ɛ-convex. Corollary 4. For any ɛ 0, a game is ɛ-convex if and only if every reliability extension of the game is ɛ-convex. 6 The converse art of Theorem 3 in [1] is technically incorrect and only holds when the game is not totally balanced, which is again generalized by our results.

8 8 Yoram Bachrach, Ian Kash, and Nisarg Shah 4.3 Relation between convexity and total balancedness Shaley [23] roved that convex games are totally balanced. In the above results, we deal with the notions of ɛ-convexity and ɛ-total balancedness. We now rove a relation between the two concets for any ɛ 0, extending Shaley s result. Theorem 3. For any ɛ 0, an ɛ/(n 1)-convex game with n agents is ɛ-totally balanced. The roof of this theorem is along the same lines as the roof of Shaley s result (see, e.g., [7]) and aears in the full version of the aer. We also show that the sufficient condition in Theorem 3 cannot be imroved by a factor of more than n 1. The roof again aears in the full version of the aer. Lemma 2. For any ɛ 0, δ > 0 and n N, there exists a game with n agents which is ɛ + δ-convex but not ɛ-totally balanced. There are several imlications of this relation. First, Corollary 4 showed that any reliability extension of an ɛ-convex game is ɛ-convex. Using Theorem 3, we can see that such an extension is also ɛ (n 1)-totally balanced. Second, the core has been well studied in the literature. For simle games, the core is non-emty if and only if a veto agent exists. In general games, convexity serves as a sufficient condition for non-emtiness of the core. However, conditions for non-emtiness of the aroximate core are relatively less studied. Theorem 3 rovides such a sufficient condition in terms of aroximate convexity. 5 Comuting a core imutation In Section 4, we roved that every reliability extension of a totally balanced game is totally balanced and thus has a non-emty core. However, the roof was non-constructive. For several classes of totally balanced games without failures, elegant LP based aroaches exist to comute a core imutation in olynomial time. But comuting a core imutation in the reliability extension may have a different comutational comlexity. For examle, Bachrach et. al. [1] note that although comuting a core imutation is easy in connectivity games on networks, even comuting the value of a coalition (and hence comuting a core imutation) becomes comutationally hard in the reliability extension. Nevertheless, we show that it is ossible to comute a core imutation in any reliability extension of a totally balanced game with high robability. In this section, we use ɛ-core for both ɛ 0 (core/aroximate core) and ɛ < 0 (suerstable core). In literature, the aroximate core is well studied in cases where the core is emty. When the core is not emty, tyically only the least core, which corresonds to the ɛ min -core (ɛ min < 0) is studied. First, we show how to comute the core (or the ɛ-core) in a reliability extension of a general game in terms of the core (or the ɛ-core) of the subgames of the base game. The latter is known to be a tractable roblem for many domains.

9 Agent Failures in Totally Balanced Games and Convex Games 9 Theorem 4. Let ɛ R. Let G = (N, v) be an ɛ-totally balanced game and G r = (N, v, r) be its reliability extension. For any coalition S N, let x S be an ɛ-core imutation of the subgame G S. Define x = S N Pr(S N) x S, where Pr(S N) = i S r i i N\S (1 r i). Then the following holds. 1. If ɛ 0, then x is an ɛ-core imutation of G r. 2. If ɛ < 0, then x is an r min ɛ-core imutation of G r, where r min = min i N r i. Proof. For every coalition S, by definition we have that x S (S) = v(s) and x S (C) v(c) ɛ for every C S. Let v r be the characteristic function of G r. First, we rove that x is a re-imutation of G r. x (N) = S N Pr(S N) x S (S) = S N Pr(S N) v(s) = v r (N), where the first transition follows since ayment to agents in N \ S is zero in x S, the second transition follows since x S (S) = v(s) and the last transition follows due to Equation (1). Now, fix any coalition C N where C N. For any C C, for all S N such that S C = C, we have x S (C) = x S (C ) v(c ) ɛ excet when S = C where we have x C (C) = x C (C ) = v(c ). Now, x (C) = Pr(S N) x S (C) = Pr(S N) x S (C) S N C C S N s.t. S C=C C C S N s.t. S C=C,S C Pr(S N) (v(c ) ɛ) + Pr(C N) v(c ) = (v(c ) ɛ) Pr(S N) + Pr(C N) ɛ C C S N s.t. S C=C = [(v(c ) ɛ) Pr(C C) + Pr(C N) ɛ] C C = [Pr(C C) v(c )] ɛ Pr(C C) Pr(C N) C C = v r (C) ɛ 1 i N\C C C (1 r i ), C C where the fourth transition follows by adding and subtracting Pr(C N) ɛ in the outer summation and rearranging terms and the last transition follows from Equation (1). Formal roofs for intuitive substitutions used in the fifth and the last transitions aear in the full version of the aer. Using this and that x is ( a re-imutation, we know that x is an ɛ -core imutation of G r if ɛ ɛ 1 ) i N\C (1 r i), for every C N such that

10 10 Yoram Bachrach, Ian Kash, and Nisarg Shah C N. If ɛ 0, then we need to maximize 1 i N\C (1 r i) else we need to minimize it. A trivial uer bound is 1 i N (1 r i) 1. We use the loose uer bound of 1. For a lower bound, note that since C N, there exists j N \ C, hence i N\C (1 r i) 1 r j 1 r min. Thus r min is a lower bound (which is also attained when C = N t where r t = r min ). This roves that x is an ɛ-core imutation of G r if ɛ 0, and an r min ɛ-core imutation if ɛ < 0. For a game G = (N, v), define ɛ as the maximum least core value over all subgames of G. That is, ɛ (G) = max S N ɛ min (G S ). Note that a subgame with a single agent has ɛ min = by definition. Thus, every subgame of G has non-emty ɛ -core and hence there exists an ɛ -core imutation for every subgame. Since an ɛ-core imutation is also an ɛ -core imutation for any ɛ ɛ (by definition), any least core imutation of any subgame of G is also an ɛ -core imutation of that subgame. Thus we obtain the following. Corollary 5. Let G, G r and r min be as defined in Theorem 4. Let ɛ denote the maximum least core value over all subgames of G. For any coalition S N, let x S be a least core imutation of G S. Define x = S N Pr(S N) x S, where Pr(S N) is as defined in Theorem 4. Then we have that 1. If ɛ 0, then x is an ɛ -core imutation of G r. 2. If ɛ < 0, then x is an r min ɛ -core imutation of G r. Consider a totally balanced game G = (N, v). Assume that we have an oracle LC such that LC(G S ) returns a least core imutation of the subgame G S of the base game G. Such an oracle subroutine exists with olynomial time comlexity for many classes of totally balanced games. For examle, Solymosi et. al. [26] give a olytime algorithm to comute the nucleolus of assignment games which are totally balanced [25]. Nucleolus is a secial (and unique) least core imutation that maximizes stability. Other examles of olytime algorithms to comute the nucleolus of totally balanced games include the algorithm by Kuiers [16] for convex games, the algorithm by Deng et. al. [9] for simle flow games and the algorithm by Kern et. al. [15] for matching games (which are totally balanced over biartite grahs [10]). For technical reasons, we extend the ayment vector LC(G S ) to a ayment vector over all agents by setting the ayments to agents in N \ S to be zero. For now, we assume that LC is deterministic, in the sense that it returns the same least core imutation every time it is called with the same subgame. We relax this assumtion in Remark 2. Observe that ɛ 0 for a totally balanced game. Corollary 5 imlies that x is an r min ɛ -core imutation of G r (and hence a core imutation as well since r min ɛ 0) that can be comuted using exonentially many calls to LC. We reduce the number of calls to LC to a olynomial in n, log(1/δ) (1 δ is the confidence level), v(n), 1/r min and 1/ ɛ by samling the subgames instead of iterating over them and using some additional tricks. However, both r min and ɛ can be exonentially small (even zero) making this algorithm ossibly an exonential time algorithm. Thus if the bound on k in Theorem 5 exceeds 2 n or if ɛ = 0, we revert to the naïve exonential summation of Corollary 5. Section 7

11 Agent Failures in Totally Balanced Games and Convex Games 11 discusses the issues with comutation of ɛ. Note that even when ɛ is unknown, the algorithm can be used in ractice by taking large number of samles. Also, the algorithm uses the value of v r (N) which is easy to aroximate by samling and the additive error can be taken care of as in Footnote 7. Algorithm CoreReliability: Comuting a core imutation of a reliability extension of a totally balanced game. Inut: Totally balanced game G = (N, v), subroutine LC to comute a least core imutation of subgames of G, reliability vector r, v r (N), δ and k. Outut: ˆx, which is in the core of G r with robability at least 1 δ. 1. Set y = For t = 1 to k do (a) For each agent i N, set l i = 1 with robability r i and l i = 0 otherwise. (b) y = y + LC(G S ) where S = {i N l i = 1} (the survivor set). 3. Let x = y/k. 4. Return ˆx = x γ 1, where 1 = 1, 1,..., 1 and γ = 1 n (x(n) vr (N)). Theorem 5. The ayment vector ˆx returned by Algorithm CoreReliability is in the core of G r with robability at least 1 δ if k 2 v(n)2 n 2 log ( ) 2 n δ rmin 2 ɛ 2. Proof. Let x S = LC(G S ). In Ste 2, every S is samled with robability Pr(S N) and the value added is x S, so E[x] = S N Pr(S N) x S = x (as in Corollary 5). For any S N and for any i N, the ayment to agent i in LC(G S ) is in [0, v(n)]. Using Hoeffding s inequality, for any i N, ( Pr x i x i r min ɛ ) ( 2 e 2 k v(n) 2 rmin ɛ ) 2 2 n. 2 n Substituting the value of k, we get that this robability is at most δ/n for every i N. Taking union bound over i N, we obtain that the robability that i N, x i x i r min ɛ 2 n, (2) holds is at least 1 δ. Now we rove that ˆx is a core imutation of G r assuming Equation (2) holds. First of all, we can see that i N x i i N x i +r min ɛ /2. But using Corollary 5, we know that x is an r min ɛ -core imutation of G r and hence i N x i = v r (N). Therefore γ in Ste 4 of the algorithm follows γ = 1 n ( i N x i v r (N) ) rmin ɛ 2 n. Hence, we can see that for every i N, ˆx i = x i γ x i r min ɛ 2 n γ x i r min ɛ, (3) n where the second transition follows due to Equation (2). Hence for any C N, ˆx(C) = ˆx i ( x i r min ɛ ) x (C) r min ɛ. (4) n i C i C

12 12 Yoram Bachrach, Ian Kash, and Nisarg Shah Since x is an r min ɛ -core imutation of G r, we know that x (C) v r (C) r min ɛ = v r (C) + r min ɛ (since ɛ < 0). Substituting this into Equation (4), we get that ˆx(C) v r (C) for every C N and C N. Furthermore, ˆx(N) = x(n) n γ = v r (N) by definition of ˆx and γ. Hence ˆx is in the core of G r. Remark 1. Note that the algorithm works so long as the subroutine LC can comute an ɛ -core imutation of every subgame of the base game. The reason why we have chosen to work with the least core is that in our case ɛ < 0 and when it is ossible to comute an ɛ -core imutation of every subgame, it is usually ossible to comute a least core imutation of every subgame as well. 7 Remark 2. Initially we assumed that LC returns a fixed least core imutation for every subgame. This is because Theorem 4 only works with fixed imutations. However, it is easy to check that if LC has any distribution over the set of all least core imutations of a subgame, then the exected ayment vector returned by LC is a least core imutation of that subgame. Thus in Algorithm CoreReliability, E[x] = E[x ] is still an r min ɛ -core imutation of the reliability extension, and the algorithm still works with the same bound on k. 8 Remark 3. Note that Hoeffding s inequality is usually alied when the requirements for the result are somewhat fuzzy whereas the requirements of a core imutation are quite strict. The use of least core imutations of subgames of the base game rovides us enough margin of error to be able to use Hoeffding s inequality and still satisfy the strict constraints with high robability. 6 Failures and Non-emtiness of the Core While studying reliability extensions of totally balanced games, we saw that introducing failures in a totally balanced game without failures, and increasing failure robabilities in a totally balanced reliability game reserve total balancedness, and hence non-emtiness of the core. In this section, we outline the effect of these two oerations on three classes of games: i) general cooerative games, ii) totally balanced games, and iii) simle games. General Games: In many games introducing failures does not reserve nonemtiness of the core. Any game G that has a non-emty core but is not totally balanced is such a game since a subgame of G with an emty core is also a reliability extension of G. Introducing failures is a secial case of increasing failure robabilities where we start with failure robabilities being zero, so increasing failure robabilities also does not reserve core non-emtiness in general games. Totally Balanced Games: Our results show that for totally balanced games, both introducing failures and more generally increasing failure robabilities reserve non-emtiness of the core (in fact, they reserve total balancedness). 7 LC can also be relaced by a subroutine LC which returns an aroximate least core imutation so long as the additive error in each comonent is less than r min ɛ /n. 8 Algorithm CoreReliability can be easily adated to comute an aroximate or suerstable core imutation of any reliability game in general with high robability.

13 Agent Failures in Totally Balanced Games and Convex Games 13 Simle Games: For simle games, Bachrach et. al. [1] observe that introducing failures reserves non-emtiness of the core. To analyze increasing failure robabilities, we erformed simulations on a secial class of simle games known as weighted voting games. A weighted voting game is defined by G = (N, w, t) where N is a set of agents where each agent i N has a weight w i 0, w is the vector of these weights and t is the threshold; a coalition C with i C w i t has v(c) = 1 and v(c) = 0 otherwise. Simulations revealed the following examle where increasing failure robabilities does not reserve core non-emtiness. Examle: Consider a weighted voting game G with 5 agents with weight vector w = 4, 3, 3, 2, 1 and threshold t = 6. Consider its reliability extension G r with the reliability vector r = 0.1, 0.6, 1, 1, 0.5. G r has a non-emty core, but decreasing the reliability of agent 5 from 0.5 to 0.1 makes the core emty. While the theme that decreasing reliability increases stability does not hold strictly for simle games, it aears to hold on average, at least for weighted voting games. We erformed several simulations where we randomly generated weighted voting games with weights samled from various distributions, e.g., uniform distribution, normal distribution, exonential distribution etc. We ket the reliabilities of all the agents equal, and observed that as this uniform reliability decreases (i.e., the failure robability increases), the robability of the core being non-emty increases. We also observed the same result for the ɛ-core. 7 Discussion and Future Work We studied the reliability extension of totally balanced games. We roved that both ɛ-total balancedness and ɛ-convexity (generalizations of the resective concets) are reserved when the reliabilities decrease. We roved a relation between these classes, generalizing a result by Shaley [23] that ties convexity and total balancedness. We also roosed an algorithm to comute a core imutaiton of any reliability extension of a totally balanced game with high robability. This oens several ossibilities for future research. First, Lemma 1 shows how the reliabilities affect the characteristic function of a game and we derived some useful results about totally balanced games and convex games building on it. Lemma 1 might also have other alications, e.g., in analyzing the effect of failures on ower indices such as the Shaley value or the Banzhaf ower index. It would also be interesting to examine how the reliability extension affects the external subsidy required to maintain stability, i.e. the Cost of Stability [3,21]. Next, the number of samles required in the algorithm resented in Section 5 deends on ɛ, the maximum least core value over all subgames of the game without failures. We are unable to settle the question of comuting ɛ or obtaining a lower bound on it in olynomial time (and thus obtaining an uer bound on the number of samles required). Such an investigation may also lead to discoveries regarding the relative stabilities of different subgames of a cooerative game. Lastly, our analysis is restircted to games where only one coalition can be formed. In contrast, cooerative games with coalitional structures [17] allow multile coalitions to arise simultaneously, and are used successfully to model

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