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Rational Synthesis
Dana Fisman, Orna Kupferman, Yoad Lustig
TL;DR
The paper asks how synthesis can exploit environments made of rational agents rather than hostile ones. It formulates rational synthesis using temporal-logic objectives, system implementations, and equilibrium strategy profiles, extending strategy logic to histories and finite-lattice payoffs. The resulting LTL rational-synthesis problem is 2EXPTIME-complete for dominant-strategy, Nash, and subgame-perfect equilibria.
Problem
Classical synthesis assumes a hostile environment, although interacting agents may pursue their own objectives; rational synthesis asks whether their rationality can be exploited while preserving the system specification.
Method
The paper outputs a system and agent strategy profile satisfying the system objective and an equilibrium, using history-extended strategy logic and extending the framework to finite distributive De-Morgan lattices.
Results
2EXPTIME-complete complexity is established for LTL rational synthesis with dominant-strategy, Nash, and subgame-perfect equilibria.
Takeaways & Limitations
Rational synthesis has the same complexity as traditional synthesis for the three studied solution concepts, and the framework supports multi-valued objectives over finite lattices.
Takeaways & Limitations
The paper restricts its quantitative rational-synthesis treatment to finite De-Morgan lattices because the full quantitative setting is undecidable in model checking.
Abstract
from arXiv · showhide
Synthesis is the automated construction of a system from its specification. The system has to satisfy its specification in all possible environments. Modern systems often interact with other systems, or agents. Many times these agents have objectives of their own, other than to fail the system. Thus, it makes sense to model system environments not as hostile, but as composed of rational agents; i.e., agents that act to achieve their own objectives. We introduce the problem of synthesis in the context of rational agents (rational synthesis, for short). The input consists of a temporal-logic formula specifying the system and temporal-logic formulas specifying the objectives of the agents. The output is an implementation T of the system and a profile of strategies, suggesting a behavior for each of the agents. The output should satisfy two conditions. First, the composition of T with the strategy profile should satisfy the specification. Second, the strategy profile should be an equilibria in the sense that, in view of their objectives, agents have no incentive to deviate from the strategies assigned to them. We solve the rational-synthesis problem for various definitions of equilibria studied in game theory. We also consider the multi-valued case in which the objectives of the system and the agents are still temporal logic formulas, but involve payoffs from a finite lattice.
1 Introduction
Rational synthesis extends automated synthesis to environments composed of rational agents with objectives of their own. It produces a system and agent strategies that satisfy the system specification while forming an equilibrium, and develops Boolean and finite-lattice solutions.
- Motivation: Traditional synthesis treats the environment as hostile, whereas rational synthesis models interacting agents as pursuing objectives other than system failure.This restricts the environment behaviors considered by synthesis.
- Problem definition: Rational synthesis outputs a system and agent strategy profile whose induced behavior satisfies the system objective and whose profile is a game-theoretic solution.The strategy profile must leave agents without incentive to deviate under the selected equilibrium concept.
- Connections: The framework relates rational synthesis to mechanism design because the system objective induces a game whose equilibrium outcome should meet a desired goal.Both system and agent objectives concern their ongoing interaction and are specified in temporal logic.
- Approach: The paper extends strategy logic with history variables to express rational synthesis for dominant strategies, Nash equilibrium, and subgame-perfect equilibrium.The extension addresses strategy logic’s inability to quantify over game histories.
- Multi-valued setting: The quantitative extension uses finite distributive De-Morgan lattices because full real-valued quantitative rational synthesis is undecidable in model checking.The lattice setting provides a decidable fragment with richer-than-Boolean payoffs.
- Related work: Related work addressed restricted rational environments, while this paper considers richer equilibrium concepts, concurrent games, LTL objectives, and the multi-valued setting.The cited prior settings include three-player games with a fair scheduler and secure equilibria, and turn-based games with parity conditions.
2 Preliminaries
The paper models infinite concurrent multiplayer games through arenas, positions, plays, and history-dependent strategies. Players choose actions simultaneously, and the transition function determines successor states.
- Game arenas: A game arena consists of nodes, an initial node, players, action sets, available-action functions, and a deterministic transition function.The transition function maps a current node and one action per player to the successor node.
- Positions and plays: A position records a game state together with one currently available action for every player.Positions combine the node component with each player’s permitted choice at that node.
- Positions and plays: A play is an infinite sequence of positions whose successive states follow the arena’s transition function.A play starting at a node begins there and must satisfy the transition relation at every step.
- Strategies: At each node, players choose actions simultaneously and independently, after which the game moves to the successor selected by the transition function.A strategy maps every finite history to an action available at the current state.
- Profiles and deviations: A strategy profile assigns one strategy to each player, while a unilateral deviation replaces only the selected player’s strategy.The notation distinguishes the original profile from the profile with Player j’s strategy replaced.
3 Rational Synthesis
Rational synthesis models a system and rational agents as a game whose strategies must both satisfy the system objective and form an accepted solution concept for the agents. The section formalizes this setting and introduces dominant strategies, Nash equilibrium, and subgame-perfect equilibrium for LTL objectives.
- Model: The world contains a system and n agents with pairwise-disjoint variable sets, and each agent has an LTL objective over all agents’ variables.Agent actions assign values to their controlled variables, while the system is treated uniformly as Agent 0.
- Model: The system strategy maps histories of actions to outputs, and rational synthesis additionally evaluates the agents’ induced game when the system follows that strategy.The induced game contains all possible behaviors of the environment agents under the system strategy.
- Problem definition: Rational synthesis returns a strategy profile whose outcome satisfies the system’s LTL formula and whose agent strategies form a solution under a selected game-theoretic concept.The supplied definition parameterizes the problem by a solution concept γ.
- Solution concepts: Dominant strategies guarantee that a player cannot lose by adhering to the assigned strategy regardless of the other players’ strategies, but some games lack such strategies.The section therefore motivates considering weaker solution concepts when dominant-strategy solutions do not exist.
- Solution concepts: Nash equilibrium prevents profitable unilateral deviation when each player assumes the others follow their assigned strategies, but it is unstable after another player deviates.In games with observed earlier choices, Nash equilibrium can also permit profiles that are not very rational because later responses may no longer be credible.
- Solution concepts: Subgame-perfect equilibrium strengthens Nash equilibrium by requiring no profitable deviation after every possible history, including histories that are not reached.For LTL objectives, the section uses Boolean payoffs: 1 when an objective holds and 0 otherwise, and defines the three concepts accordingly.
4 Solution in the Boolean Setting
The Boolean solution extends strategy logic with histories to express rational-synthesis requirements, then reduces satisfiability to alternating parity tree-automata emptiness. This yields a 2EXPTIME-complete procedure for dominant-strategy, Nash, and subgame-perfect equilibria.
- Extended Strategy Logic: Extended Strategy Logic adds history variables so specifications can evaluate outcomes from arbitrary histories, enabling subgame-perfect equilibrium reasoning.Ordinary strategy logic cannot refer to histories that diverge from the assigned strategies; ESL addresses this limitation.
- Extended Strategy Logic: ESL expresses dominant-strategy, Nash, and subgame-perfect equilibrium requirements through quantified strategy and history variables.A closed formula existentially quantifies a strategy profile, requires the system objective, and asserts the chosen solution concept.
- Automata Construction: ESL formulas are translated inductively into alternating parity tree automata that track strategy profiles, histories, and LTL satisfaction along designated paths.The automata use projection for strategy and history quantification and check the path determined by a history followed by a strategy profile.
- Automata Construction: Theorem 4.1 gives a (d + 1)-EXPTIME emptiness procedure for ESL formulas with alternation depth d.The constructed automaton accepts exactly the satisfying assignments [[Ψ]]G.
- Extensions: The same algorithm extends to rational control by incorporating the controlled system’s state space into the strategy-tree representation.In rational control, strategy trees become (S × Σ)-trees rather than Σ-trees.
5 Solution in the Multi-Valued Setting
The multi-valued setting assigns temporal-logic objectives payoffs from finite De-Morgan lattices and restricts attention to deterministic latticed Büchi automata with Nash equilibrium. A lattice-game reduction to Boolean generalized-Büchi games supports decidability and an EXPTIME upper bound.
- Problem Setting: The multi-valued extension uses finite De-Morgan lattices to represent payoffs for temporal-logic objectives.The full quantitative setting is undecidable, so the paper studies this finite-lattice fragment.
- Problem Setting: Latticed rational synthesis seeks a strategy profile giving the system payoff at least v while making the agents’ strategies a solution under a chosen equilibrium concept.Definition 5.1 formalizes these as separate payoff and equilibrium conditions.
- Automata Restriction: The algorithm restricts qualitative specifications to LLTL formulas translatable into deterministic latticed Büchi word automata and focuses on Nash equilibrium.This restriction reflects the less-developed automata-theoretic machinery for latticed specifications.
- Game Reduction: The reduction handles the fact that lattice-game values may arise by joining outcomes of different strategies, whereas a strategy tree represents one strategy.The single-strategy condition is central to the latticed tree-automata emptiness construction.
- Game Reduction: Theorem 5.2 reduces achieving lattice value at least l with one strategy to winning a Boolean generalized-Büchi game Gl.The constructed game has size bounded by |G| · |Ł|^2 and G1 has at most |Ł| acceptance sets.
- Complexity: Theorem 5.3 places latticed rational synthesis for LDBW objectives and Nash equilibrium in EXPTIME.The apparently lower complexity than the Boolean setting reflects that LDBW objectives are less succinct than LLTL formulas.
6 Discussion
Rational synthesis separates system correctness from agents’ incentives by suggesting strategies that rational agents have no reason to abandon. Its extended strategy-logic framework supports several equilibrium concepts and extensions such as robust specifications.
- Core Perspective: Rational synthesis combines a system objective with agent objectives and assumes agents follow suggested strategies when deviation is not worthwhile.Unlike traditional synthesis, the environment is modeled through objectives and equilibrium behavior rather than only universal adversarial choices.
- Methodological Contribution: The extended strategy-logic solution separates game-theoretic solution concepts from the synthesis technique.This modularity allows the technique to apply to any solution concept expressible in extended strategy logic.
- Supported Equilibria: For dominant strategies, Nash equilibrium, and subgame-perfect equilibrium, rational synthesis has the same complexity as traditional synthesis.The framework also permits different solution concepts for different subspecifications.
- Extensions: The framework can require additional system properties to hold in all environments, independently of agents’ rationality.This is expressed by adding an LTL formula alongside the rational-synthesis objective.
A.1 Proof of Theorem 4.1
The proof constructs an alternating parity tree automaton inductively over ESL formulas and analyzes its size and emptiness complexity. For base formulas, the automaton checks legal history labeling, follows the strategy from the history, and verifies the LTL formula.
- The automaton construction proceeds by induction on the structure of Ψ, using closure under union, complementation, and projection for Boolean and quantified cases.
- For alternation depth d, AΨ has (d + 1)-exponential state complexity and polynomial index of degree d, with projection avoiding an additional exponential emptiness blow-up.
- For a base formula ψ(z, h), AΨ checks legal history labeling, follows the strategy profile from h, and verifies that the resulting path satisfies ψ.The construction uses history and future states to detect strategy violations and illegal labeling.
- The constructed automaton has exponentially many states in the formula and index 3 for the base construction.
- A tree is accepted exactly when some finite history path is marked by ⊤ labels and the corresponding outcome satisfies ψ.
A.2 Proof of Theorem 5.2
The proof reduces latticed games to Boolean generalized-Büchi games by tracking relinquished values, then establishes equivalence of winning strategies and value guarantees.
- A strategy guarantees value at least l exactly when it guarantees at least every join-irreducible element below l.This uses Birkhoff’s representation theorem and defines X_l as the relevant join-irreducible elements.
- The Boolean reduction expands each latticed-game state into a tuple containing the original state and two tracked relinquished values.
- For each join-irreducible x below l, the generalized-Büchi condition requires infinitely frequent collection of x through relinquishment or acceptance.
- A strategy in the latticed game induces a winning strategy in the Boolean reduction because every required join-irreducible value is collected infinitely often.
- Conversely, a winning strategy in the Boolean reduction induces a latticed-game strategy whose play has value at least l.
A.3 Proof of Theorem 5.3
The proof constructs tree automata for system payoff and Nash equilibria, intersects them, and checks the resulting Streett automaton for emptiness.
- The automaton A0 accepts strategy profiles whose system payoff is at least the target value, while AN accepts Nash equilibria; their intersection contains the solutions.
- The construction uses lattice size m and agent specification sizes si, with s denoting the maximum specification size.
- A0 is obtained from a generalized-Büchi tree automaton of size s0 · m^2 with at most m acceptance sets.
- Nash-equilibrium automata compare each agent’s payoff under a deviation with the payoff under the proposed profile for every join-irreducible lattice value.
- The automaton AN has size (s · m)^O(n) and O(m · n) Streett pairs.
- The final intersection is a Streett automaton of size (s · m)^O(n) with O(m · n) pairs, whose emptiness is checked in time (s · m)^O(m · n^2).