# Boolean Games: Inferring Agents' Goals Using Taxation Queries

IJCAI 2020, pp. 1585-1591, 2020.

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Abstract:

In Boolean games, each agent controls a set of Boolean variables
and has a goal represented by a propositional formula. We study
inference problems in Boolean games assuming the presence of a
PRINCIPAL who has the ability to control the agents and impose
taxation schemes. Previous work used taxation schemes to guide a
game towards...More

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Introduction

- Boolean games [Harrenstein et al, 2001] are a class of games where agents’ goals are represented by a propositional logic formula.
- Each agent i controls a distinct set of Boolean variables Φi, and there is a cost associated with each assignment.
- Much of the work on Boolean games is of theoretical nature [Harrenstein et al, 2001; Sauro and Villata, 2013; Wooldridge et al, 2013; Grant et al, 2014; Agotnes et al, 2013; Bonzon et al, 2007; Levit et al, 2013b].
- Boolean games have been used to model some real-world problems such as charging electric vehicles and traffic signalling [Levit et al, 2013b]

Highlights

- Boolean games [Harrenstein et al, 2001] are a class of games where agents’ goals are represented by a propositional logic formula
- When there is an Nash Equilibrium for a {0,1}-taxation query, we show that the value of the goals of selected agents for the zero cost assignment can be determined regardless of which Nash Equilibrium is reached by the agents (Section 3)
- We show experimentally that our coloring-based inference algorithm uses significantly fewer queries compared to inferring goals one at a time
- We show that goal inference is possible in a context where the response to a query only provides the values assigned to the variables and not whether agents achieved their goals
- Our algorithms reduce the number of taxation queries by querying many agents simultaneously
- We ran an extensive set of experiments creating a large number of networks and agents’ goals by varying the number of nodes, the minimum degree, the number of variables in an agent’s goal controlled by the agent and by others and the threshold

Results

- The authors show experimentally that the coloring-based inference algorithm uses significantly fewer queries compared to inferring goals one at a time.
- The coloring-based approach uses significantly less time even for games with 36,000 agents (Section 5).
- The authors use SEQ to denote the querying algorithm that infers goals one node at a time and CBQ to denote the coloring-based querying algorithm.
- Nodes represent agents and the network itself represents the goal overlap graph
- The authors ensured this by creating for each edge {u, v}, a variable xu,v that appears only in the goals of u and v.
- The authors generated a random threshold value in the range [1 .. |Γi|]

Conclusion

- Necessary and sufficient conditions for the existence of an NE for a taxation query.
- Using an undirected graph that captures the overlaps between the sets of variables used in agents’ goals, the authors establish necessary and sufficient conditions for the existence of an NE for a Boolean game and any {0,1}-taxation query, i.e., a query with only 0 and 1 costs (Section 3).

Summary

## Introduction:

Boolean games [Harrenstein et al, 2001] are a class of games where agents’ goals are represented by a propositional logic formula.- Each agent i controls a distinct set of Boolean variables Φi, and there is a cost associated with each assignment.
- Much of the work on Boolean games is of theoretical nature [Harrenstein et al, 2001; Sauro and Villata, 2013; Wooldridge et al, 2013; Grant et al, 2014; Agotnes et al, 2013; Bonzon et al, 2007; Levit et al, 2013b].
- Boolean games have been used to model some real-world problems such as charging electric vehicles and traffic signalling [Levit et al, 2013b]
## Results:

The authors show experimentally that the coloring-based inference algorithm uses significantly fewer queries compared to inferring goals one at a time.- The coloring-based approach uses significantly less time even for games with 36,000 agents (Section 5).
- The authors use SEQ to denote the querying algorithm that infers goals one node at a time and CBQ to denote the coloring-based querying algorithm.
- Nodes represent agents and the network itself represents the goal overlap graph
- The authors ensured this by creating for each edge {u, v}, a variable xu,v that appears only in the goals of u and v.
- The authors generated a random threshold value in the range [1 .. |Γi|]
## Conclusion:

Necessary and sufficient conditions for the existence of an NE for a taxation query.- Using an undirected graph that captures the overlaps between the sets of variables used in agents’ goals, the authors establish necessary and sufficient conditions for the existence of an NE for a Boolean game and any {0,1}-taxation query, i.e., a query with only 0 and 1 costs (Section 3).

- Table1: A game without a NE for some taxation schemes

Related work

- Harrenstein et al [2001] introduced Boolean games as a class of two-player games and Bonzon et al [2007] generalized the framework to n players. Structural and computational properties of PNE in Boolean games are well-studied. Bonzon et al [2007] define a “dependency graph” between players to characterize PNE in Boolean games, much like our goal overlap graph. Levit et al [2019; 2013b] study methods for finding a taxation scheme that incentivizes the agents to reach a stable state. Also, Levit et al [2013a] discuss an application of Boolean games to the charging of electric vehicles where some vehicles are not allowed to charge at certain time intervals to avoid overloading. This is similar to our notion of the PRINCIPAL inhibiting agents. Boolean games where players have incomplete information about each other’s goals have also been considered in the literature (e.g., [Clercq et al, 2015; Agotnes et al, 2013]).

Funding

- This work was partially supported by NSF Grants ACI-1443054 (DIBBS), IIS-1633028 (BIG DATA), CMMI-1745207 (EAGER), OAC-1916805, IIS-1908530 and by the Ministry of Science & Technology, Israel and the Ministry of Education, Science, Research and Sport of the Slovak Republic

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