The Model Zero Knowledge Arbitrary Knowledge Full Knowledge Selfish Load Balancing under Partial Knowledge Panagiota N. Panagopoulou Research Academic Computer Technology Institute University of Patras, Greece panagopp@cti.gr joint work with Elias Koutsoupias and Paul Spirakis AEOLUS Workshop on Scheduling Nice, March 8–9, 2007 Panagiota N. Panagopoulou Selfish Load Balancing under Partial Knowledge
The Model Zero Knowledge Arbitrary Knowledge Full Knowledge Outline The Model 1 Agents and strategies Selfish Costs and Nash equilibria The Divergence Ratio Zero Knowledge 2 All Nash equilibria The Divergence Ratio Arbitrary Knowledge 3 Bounding the Players’ Optimum Bounding the worst Social Cost A lower bound on the Divergence Ratio Full Knowledge 4 The Divergence Ratio Panagiota N. Panagopoulou Selfish Load Balancing under Partial Knowledge
The Model Agents and strategies Zero Knowledge Selfish Costs and Nash equilibria Arbitrary Knowledge The Divergence Ratio Full Knowledge Outline The Model 1 Agents and strategies Selfish Costs and Nash equilibria The Divergence Ratio Zero Knowledge 2 All Nash equilibria The Divergence Ratio Arbitrary Knowledge 3 Bounding the Players’ Optimum Bounding the worst Social Cost A lower bound on the Divergence Ratio Full Knowledge 4 The Divergence Ratio Panagiota N. Panagopoulou Selfish Load Balancing under Partial Knowledge
The Model Agents and strategies Zero Knowledge Selfish Costs and Nash equilibria Arbitrary Knowledge The Divergence Ratio Full Knowledge Agents, loads and information A set N = { 1 , 2 , . . . , n } of n > 1 selfish agents Each i ∈ N has a load w i ∈ [0 , 1] Two bins (bin 0 and bin 1) of unbounded capacity Each agent has to select one of the two available bins to put her load For any ( i , j ) ∈ N × N , agent i knows either (a) the exact value of w j or (b) that w j is uniformly distributed on [0 , 1] Let I i = { j ∈ N : agent i knows the exact value of w j } and denote I = ( I i ) i ∈ N Panagiota N. Panagopoulou Selfish Load Balancing under Partial Knowledge
The Model Agents and strategies Zero Knowledge Selfish Costs and Nash equilibria Arbitrary Knowledge The Divergence Ratio Full Knowledge Single-threshold strategies A strategy for agent i is a function s i : [0 , 1] → { 0 , 1 } such that s i ( w i ) is the bin that agent i selects when her load is w i . We only consider single-threshold strategies, i.e. a strategy for agent i is some t i ∈ [0 , 1] so that � 0 w i ≤ t i s i ( w i ) = 1 w i > t i A strategy profile t = ( t 1 , . . . , t n ) ∈ [0 , 1] n is a combination of strategies, one for each agent. Denote by ( t ′ i , t − i ) the strategy profile that is identical to t except for agent i , who chooses strategy t ′ i instead of t i . Panagiota N. Panagopoulou Selfish Load Balancing under Partial Knowledge
The Model Agents and strategies Zero Knowledge Selfish Costs and Nash equilibria Arbitrary Knowledge The Divergence Ratio Full Knowledge Selfish Costs and Nash equilibria The Selfish Cost Cost i ( t ; I i ) of agent i is the expected load of the bin she selects, based on 1 her information about the exact loads of all j ∈ I i and 2 her knowledge that the loads of all j / ∈ I i are uniformly distributed on [0 , 1]. Panagiota N. Panagopoulou Selfish Load Balancing under Partial Knowledge
The Model Agents and strategies Zero Knowledge Selfish Costs and Nash equilibria Arbitrary Knowledge The Divergence Ratio Full Knowledge Selfish Costs and Nash equilibria The Selfish Cost Cost i ( t ; I i ) of agent i is the expected load of the bin she selects, based on 1 her information about the exact loads of all j ∈ I i and 2 her knowledge that the loads of all j / ∈ I i are uniformly distributed on [0 , 1]. In a Nash equilibrium , no agent can decrease her Selfish Cost by deviating: Definition The strategy profile t = ( t 1 , . . . , t n ) ∈ [0 , 1] n is a Nash equilibrium if and only if, for all i ∈ N , ( t ′ ∀ t ′ � � Cost i ( t ; I i ) ≤ Cost i i ∈ [0 , 1] . i , t − i ); I i Panagiota N. Panagopoulou Selfish Load Balancing under Partial Knowledge
The Model Agents and strategies Zero Knowledge Selfish Costs and Nash equilibria Arbitrary Knowledge The Divergence Ratio Full Knowledge The Divergence Ratio Associated with a strategy profile t ∈ [0 , 1] n is the Social Cost : SC( t , I ) = max i ∈ N Cost i ( t ; I i ) . The Players’ Optimum PO( I ) is the minimum, over all possible strategy profiles t ∈ [0 , 1] n , Social Cost: PO( I ) = t ∈ [0 , 1] n SC( t , I ) . min The Divergence Ratio DR( I ) is the maximum, over all Nash equilibria t , of the ratio SC( t , I ) PO( I ) : SC( t , I ) DR( I ) = max . PO( I ) t : t N.E. Panagiota N. Panagopoulou Selfish Load Balancing under Partial Knowledge
The Model Zero Knowledge All Nash equilibria Arbitrary Knowledge The Divergence Ratio Full Knowledge Outline The Model 1 Agents and strategies Selfish Costs and Nash equilibria The Divergence Ratio Zero Knowledge 2 All Nash equilibria The Divergence Ratio Arbitrary Knowledge 3 Bounding the Players’ Optimum Bounding the worst Social Cost A lower bound on the Divergence Ratio Full Knowledge 4 The Divergence Ratio Panagiota N. Panagopoulou Selfish Load Balancing under Partial Knowledge
The Model Zero Knowledge All Nash equilibria Arbitrary Knowledge The Divergence Ratio Full Knowledge Zero Knowledge Assume I i = ∅ for all i ∈ N . Then Cost i ( t ; I i ) = Panagiota N. Panagopoulou Selfish Load Balancing under Partial Knowledge
The Model Zero Knowledge All Nash equilibria Arbitrary Knowledge The Divergence Ratio Full Knowledge Zero Knowledge Assume I i = ∅ for all i ∈ N . Then t i t j � Cost i ( t ; I i ) = t i 2 + t j 2 j � = i Panagiota N. Panagopoulou Selfish Load Balancing under Partial Knowledge
The Model Zero Knowledge All Nash equilibria Arbitrary Knowledge The Divergence Ratio Full Knowledge Zero Knowledge Assume I i = ∅ for all i ∈ N . Then t i t j � Cost i ( t ; I i ) = t i 2 + t j 2 j � = i t i + 1 (1 − t j ) t j + 1 � +(1 − t i ) + 2 2 j � = i Panagiota N. Panagopoulou Selfish Load Balancing under Partial Knowledge
The Model Zero Knowledge All Nash equilibria Arbitrary Knowledge The Divergence Ratio Full Knowledge Zero Knowledge Assume I i = ∅ for all i ∈ N . Then t i t j � Cost i ( t ; I i ) = t i 2 + t j 2 j � = i t i + 1 (1 − t j ) t j + 1 � +(1 − t i ) + 2 2 j � = i Proposition j − n − 1 2 − 1 + n � t 2 � t 2 Cost i ( t ; I i ) = t i j . 2 2 j � = i j � = i Panagiota N. Panagopoulou Selfish Load Balancing under Partial Knowledge
The Model Zero Knowledge All Nash equilibria Arbitrary Knowledge The Divergence Ratio Full Knowledge Characterization of Nash equilibria �� � j � = i t 2 j − n − 1 + n 2 − 1 j � = i t 2 � Since Cost i ( t ; I i ) = t i j : 2 2 Proposition Consider the case where I i = ∅ for all i ∈ N. Then the strategy profile t ∈ [0 , 1] n is a Nash equilibrium if and only if, for all i ∈ N, j ≥ n − 1 � t 2 t i = 0 ⇒ 2 j � = i j ≤ n − 1 � t 2 ⇒ t i = 1 2 j � = i j = n − 1 � t 2 t i ∈ (0 , 1) ⇒ 2 j � = i Panagiota N. Panagopoulou Selfish Load Balancing under Partial Knowledge
The Model Zero Knowledge All Nash equilibria Arbitrary Knowledge The Divergence Ratio Full Knowledge All Nash equilibria Theorem Consider the case where I i = ∅ for all i ∈ N. Then the strategy profile t ∈ [0 , 1] n is a Nash equilibrium if and only if κ agents choose threshold 1 , λ agents choose threshold t A ∈ (0 , 1) , n − κ − λ agents choose threshold 0 and n − 1 − λ ≤ κ ≤ n − 1 2 , λ > 1 , t 2 n − 1 κ (1) A = 2( λ − 1) − λ − 1 or 2 (2) n is even, κ = n 2 , λ = 0 or (3) n is odd, κ = n +1 2 , λ = 0 or (4) n is odd, κ = n − 1 2 , λ = 0 or (5) n is odd, κ = n − 1 2 , λ = 1 . The maximum, over all Nash equilibria, Social Cost is n +1 4 . Panagiota N. Panagopoulou Selfish Load Balancing under Partial Knowledge
The Model Zero Knowledge All Nash equilibria Arbitrary Knowledge The Divergence Ratio Full Knowledge All Nash equilibria Sketch of Proof. In order to find all Nash equilibria we have to find all the possible partitions of the set of agents into three sets A , B and C so that For all i ∈ A , t i = t A for some t A ∈ (0 , 1) and A + | C | = n − 1 ( | A | − 1) t 2 2 . For all i ∈ B , t i = 0 and | A | t 2 A + | C | ≥ n − 1 2 . A + | C | − 1 ≤ n − 1 For all i ∈ C , t i = 1 and | A | t 2 2 . We consider the cases | A | = 0 , | A | = 1 and | A | > 1 so as to find all Nash equilibria and calculate their Social Cost. Panagiota N. Panagopoulou Selfish Load Balancing under Partial Knowledge
The Model Zero Knowledge All Nash equilibria Arbitrary Knowledge The Divergence Ratio Full Knowledge The Divergence Ratio Lemma PO( I ) = n 4 if n is even and PO( I ) = n +1 if n is odd. 4 Therefore Theorem Consider the case where I i = ∅ for all i ∈ N. Then DR( I ) = 1 + 1 n if n is even and DR( I ) = 1 if n is odd. Panagiota N. Panagopoulou Selfish Load Balancing under Partial Knowledge
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