# WP3

## WP3: Strategic Models for Collective Behaviour

**Coordinators: Laurent Gourvès and Matias Nuñez**

This work package mainly deals with the impact of strategic behaviour on the feasibility and the quality of the collective decision mechanisms. It first deals with game-theoretic solution concepts in voting contexts. It also focuses on the notion of rationality and its role in voting environments (alternative notions of rationality). It finally deals with (in)efficiencies that can be generated by the different notions of rationality.

**Alternative solution concepts: existence, convergence, computation**

As far as solutions concepts are concerned, it is clear that some of them are more attractive than others in the games formed by agents reporting their preferences in collective decision mechanisms. For instance, Nash equilibria in standard voting contexts are often too numerous, and almost any outcome is possible in equilibrium, so that this concept has a poor predictive power. Its extensions, such as correlated equilibria (CE) and refinements such as strong equilibria (SE) are appealing concepts that are particularly relevant in collective decision contexts. Likewise, dominance solvability, through the iterated elimination of weakly dominated strategies (IEWDS), has recently been used in strategic voting contexts since it gives a neutral measure of the coordination incentives that each voting rule posits. We explore these solution concepts. In particular, we address the following questions: in which collective decision games a CE, SE is guaranteed to exist, or when is a game solvable through IEWDS? If the answer is positive, is it computationally hard to compute such an equilibrium? After a sequence of improving deviations made by the players (each one at a time), do the players reach an equilibrium? If an equilibrium is reached, what it the rate of convergence?

Potential applications of this research stream include electoral behaviour. Indeed, the equilibrium dynamics can be useful to describe how the information impacts the (strategic) voters’ choices. For instance, in the French presidential election (plurality with runoff), it is far from clear how the voters should vote in the first round. This might trigger coordination failures and poor preference/ information aggregation. Can we modify the electoral system so that the voters’ opinions are correctly aggregated in equilibrium? The experimental work (see WP4) focuses on these “opinion dynamics” and complements our theoretical findings.

**Beyond standard notions of rationality**

Rationality is a basic axiom which is often interpreted as saying that players are self-interested. However agents (a fortiori, human beings) are sometimes not fully rational, at least, not according to the most standard definition of rationality, and it is of paramount importance to understand and formalize this phenomenon. A first interpretation is that the players’ actions are affected by fairness, altruism or even spite. A player may optimize a combination of individual cost and social cost. A second interpretation is that a deviation is performed only if it induces a “significant” modification of the player’s utility. Finally, the idea of partial honesty is also a relevant alternative to the usual postulate of full rationality. A partially honest player has a strict preference for telling the truth only when truth-telling leads to an outcome which is not worse than the outcome which occurs when he lies. We revisit the collective decision games in which a player’s action is guided by one of the above alternative notions of rationality: existence, computation and dynamics of the equilibria associated with these notions of rationality. The recent theory of voting has mostly focused on environments with a large number of voters (as in national elections). In these environments, characterizing several properties of preference/information aggregation becomes more tractable. In contrast with this strand of the literature, we mostly focus on elections with few voters. Indeed, one can understand voting with few agents as a specific sort of bargaining that does not fit into the usual Nash approach.

**How bad is the game's outcome ?**

The last topic is the performance degradation of systems due to the selfish behavior of the users. The price of anarchy and the price of stability (abbreviated PoA/PoS) compare the social cost of the worst/best Nash equilibrium with the social optimum, in the worst case. The PoA/PoS were originally defined for Nash equilibria but it is natural to extend them to other solution concepts. We study the PoA/PoS for refinements of the Nash equilibrium. Besides self-interest, spite and more surprisingly altruism can lead to socially poor outcomes. Being fair may also harm the system’s performance. Another goal is to bound the PoA/PoS when the players follow alternative notions of rationality. To what extent, following an alternative notion of rationality can improve or deteriorate the quality of the system. A similar question is posed in the context of mechanism design. Returning the socially optimal outcome is rarely strategyproof. Therefore one has to sacrifice optimality to ensure strategyproofness.