# Self-stabilizing Uncoupled Dynamics

SAGT, pp. 74-85, 2014.

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Abstract:

Dynamics in a distributed system are self-stabilizing if they are guaranteed to reach a stable state regardless of how the system is initialized. Game dynamics are uncoupled if each player's behavior is independent of the other players' preferences. Recognizing an equilibrium in this setting is a distributed computational task. Self-sta...More

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Introduction

- Self-stabilization is a failure-resilience property that is central to distributed computing theory and is the subject of extensive research [3].
- The authors prove that historyless deterministic uncoupled dynamics cannot succeed on all games over any action profile space (Theorem 18).
- The authors determine, for every profile space A, the minimum r ∈ N such that an uncoupled r-recall stationary strategy mapping can succeed on all games (A, U ) ∈ (A, U(A)) or all generic games (A, U ) ∈ (A, G(A)).

Highlights

- Self-stabilization is a failure-resilience property that is central to distributed computing theory and is the subject of extensive research [3]. It is characterized by the ability of a distributed system to reach a stable state from every initial state
- The dynamics we study will have bounded recall, so the state of this system at any time consists of the r most recent action profiles, for some finite r
- Our results We show in Section 3 that historyless uncoupled dynamics can succeed on all two-player games with a two-action player and on all three-player generic games with a two-action player (Theorems 6 and 11)
- We prove that historyless deterministic uncoupled dynamics cannot succeed on all games over any action profile space (Theorem 18)
- Continuing inductively, since σ is cyclic, unless the players converge to a pure Nash equilibrium, they will examine every profile v ∈ A with a query state of the form (u, v)

Results

- Theorem 1 (Hart and Mas-Colell [7]) For any profile space A, there exists an uncoupled 2-recall stationary strategy mapping that succeeds on all games (A, U ).
- Theorem 2 (Hart and Mas-Colell [7]) There is no uncoupled historyless stationary strategy mapping that succeeds on all three-player games, or on all 3-by3-by-3 generic games.
- Theorem 3 (Hart and Mas-Colell [7]) For any two-player profile space A, there is an uncoupled historyless stationary strategy mapping that succeeds on all games (A, U ).
- Definition For any n-player profile space A, the canonical historyless uncoupled stationary strategy mapping for A is h : U(A) → F(A), defined as follows.
- Unless A has only two players and one of those players has only two actions, no historyless uncoupled strategy mapping succeeds on all games (A, U ).
- Lemma 8 says that additional actions do not make a profile space any “easier” in this context; the players will need at least as much recall to succeed on all games in the larger space.
- If A has more than three players, or if every player has more than two actions, no uncoupled historyless stationary strategy mapping can succeed on all generic games (A, U ).
- Theorem 16 For every profile space A, there exists a deterministic uncoupled 3-recall stationary strategy mapping that succeeds on all games (A, U ).
- Theorem 17 If A is a profile space in which every player has at least four actions, there exists a 2-recall deterministic uncoupled stationary strategy mapping that succeeds on all games (A, U ).

Conclusion

- Continuing inductively, since σ is cyclic, unless the players converge to a PNE, they will examine every profile v ∈ A with a query state of the form (u, v).
- While deterministic uncoupled 2-recall dynamics can succeed on at least some classes that require 2-recall in the stochastic setting, historyless dynamics of this type fail on U(A) for every profile space A.
- Theorem 18 For every profile space A, no deterministic uncoupled historyless stationary strategy mapping succeeds on all games (A, U ).

Summary

- Self-stabilization is a failure-resilience property that is central to distributed computing theory and is the subject of extensive research [3].
- The authors prove that historyless deterministic uncoupled dynamics cannot succeed on all games over any action profile space (Theorem 18).
- The authors determine, for every profile space A, the minimum r ∈ N such that an uncoupled r-recall stationary strategy mapping can succeed on all games (A, U ) ∈ (A, U(A)) or all generic games (A, U ) ∈ (A, G(A)).
- Theorem 1 (Hart and Mas-Colell [7]) For any profile space A, there exists an uncoupled 2-recall stationary strategy mapping that succeeds on all games (A, U ).
- Theorem 2 (Hart and Mas-Colell [7]) There is no uncoupled historyless stationary strategy mapping that succeeds on all three-player games, or on all 3-by3-by-3 generic games.
- Theorem 3 (Hart and Mas-Colell [7]) For any two-player profile space A, there is an uncoupled historyless stationary strategy mapping that succeeds on all games (A, U ).
- Definition For any n-player profile space A, the canonical historyless uncoupled stationary strategy mapping for A is h : U(A) → F(A), defined as follows.
- Unless A has only two players and one of those players has only two actions, no historyless uncoupled strategy mapping succeeds on all games (A, U ).
- Lemma 8 says that additional actions do not make a profile space any “easier” in this context; the players will need at least as much recall to succeed on all games in the larger space.
- If A has more than three players, or if every player has more than two actions, no uncoupled historyless stationary strategy mapping can succeed on all generic games (A, U ).
- Theorem 16 For every profile space A, there exists a deterministic uncoupled 3-recall stationary strategy mapping that succeeds on all games (A, U ).
- Theorem 17 If A is a profile space in which every player has at least four actions, there exists a 2-recall deterministic uncoupled stationary strategy mapping that succeeds on all games (A, U ).
- Continuing inductively, since σ is cyclic, unless the players converge to a PNE, they will examine every profile v ∈ A with a query state of the form (u, v).
- While deterministic uncoupled 2-recall dynamics can succeed on at least some classes that require 2-recall in the stochastic setting, historyless dynamics of this type fail on U(A) for every profile space A.
- Theorem 18 For every profile space A, no deterministic uncoupled historyless stationary strategy mapping succeeds on all games (A, U ).

Related work

- In addition to the results mentioned above, Hart and Mas-Colell also addressed convergence to mixed Nash equilibria by bounded-recall uncoupled dynamics [7]. Babichenko investigated the situation when the uncoupled players are finite-state automata, as well as completely uncoupled dynamics, in which each player can see only the history of its own actions and payoffs [1,2]. Hart and Mansour [5] analyzed the time to convergence for uncoupled dynamics. Jaggard, Schapira, and Wright [9] investigated convergence to pure Nash equilibria by game dynamics that are distributed in the sense of being asynchronous, rather than uncoupled.

Games Let n ∈ N and (k1, ..., kn) ∈ Nn, with each ki ≥ 2. A game of size (k1, ..., kn) is a pair (A, U ), where A = A1 × ... × An such that each |Ai| = ki, and U = (u1, ..., un) is an n-tuple of functions ui : A → R. Ai and ui are the action set and utility function of player i. When n is small, we may describe a game (A, U ) as a k1-by-...-by-kn game. Elements of A are the (action) profiles of the game, and A is called the (action) profile space. U(A) is the the class of all U such that each ui takes Ai as input, so (A, U (A)) is the class of all games with profile space A. When A is clear from context, we often identify the game with the utility function vector U .

Reference

- Y. Babichenko. Uncoupled automata and pure Nash equilibria. Int. J. Game Theory, 39(3):483–502, 2010.
- Y. Babichenko. Completely uncoupled dynamics and Nash equilibria. Games and Economic Behavior, 76(1):1–14, 2012.
- S. Dolev. Self-Stabilization. MIT Press, 2000.
- D. Fudenberg and J. Tirole. Game theory (3. pr.). MIT Press, 1991.
- S. Hart and Y. Mansour. How long to equilibrium? The communication complexity of uncoupled equilibrium procedures. Games and Economic Behavior, 69(1):107– 126, 2010.
- S. Hart and A. Mas-Colell. Uncoupled dynamics do not lead to Nash equilibrium. American Economic Review, 93(5):1830–1836, 2003.
- S. Hart and A. Mas-Colell. Stochastic uncoupled dynamics and Nash equilibrium. Games and Economic Behavior, 57(2):286–303, 2006.
- S. Hart and A. Mas-Colell. Simple Adaptive Strategies: From Regret-matching to Uncoupled Dynamics. World Scientific Publishing Co., Inc., River Edge, NJ, USA, 2012.
- A. D. Jaggard, M. Schapira, and R. N. Wright. Distributed computing with adaptive heuristics. In B. Chazelle, editor, ICS, pages 417–443. Tsinghua University Press, 2011.
- N. Nisan, T. Roughgarden, E. Tardos, and V. V. Vazirani. Algorithmic Game Theory. Cambridge University Press, New York, NY, USA, 2007.
- 2. It follows that the players will never reach the PNE (3, 3, 3) starting from, for example, (1, 2, 1), when following hU.

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