# Rational Synthesis

## TL;DR

This paper introduces rational synthesis, a method for automatically constructing systems interacting with rational agents, ensuring the system's specifications are met and agents have no incentive to deviate, based on game-theoretic equilibria.

## Contribution

It formalizes the rational synthesis problem, integrating game theory with system synthesis, and provides solutions for various equilibrium concepts and multi-valued objectives.

## Key findings

- Solved rational-synthesis for multiple equilibrium definitions.
- Extended the framework to multi-valued objectives with finite lattice payoffs.
- Demonstrated the approach's effectiveness through theoretical results.

## Abstract

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.