In this work we will present and discuss some simulative and analytical results on the emergence of social conventions. We claim that a convention (Gilbert 1981; Lewis 1969; Sugden, 1986/2004; Young 1993) is a behavioural regularity. More specifically conventions are a class of problems classified as coordination games (the convention of keeping to the right (or left) when driving, pointed out by David Lewis, 1969), based on interdependency and mutual expectations. One of the main purposes of conventions is to coordinate people's expectations in socio-economic interactions that have multiple equilibria. In particular, our study focuses on the emergence of conventions in (traffic) interactions, drawing on Sen and Airiau’s study (2007) of the emergence of a precedence rule in the traffic. In our model the world is reticular and toroidal. The agents' population presents four different strategies; each of them is shown by the relative agents' numerousness (if N is the number of agents and each strategy has the same numerousness, each subpopulation consists of N/4 agents). Each subpopulation shows a different behaviour when an agent reaches a grid's node; we assume that (i) on each node can stay maximum four agents and (ii) each agent must have a different direction (0°, 90°, 180°, 270°). The strategies adopted are: 1. cross the node; 2. cross the node only if there are not agents coming from both perpendicular directions (+90° and -90°); 3. cross the node only if there is not an agent coming from the first of the perpendicular directions (+90°); 4. cross the node only if there is not an agent coming from the second of the perpendicular directions (-90°). The agents' payoff is the result of node crossing: the payoff is 1, if the agent avoids collisions; -1, otherwise; the payoff is 0 if there are not agents coming from +90° and -90°. We implement two different processes of imitation learning: local and global. In the first case (local) an agent imitates the strategy of the agent which reached the best payoff in the same node; in the second (global), an agent imitates the strategy of another agent which reached the best payoff in another node. We realized an analytical model of all possible interactions in our scenario. We calculated some point of equilibrium (Nash equilibrium); we found that the results obtained are very close to the agent-based simulations' results.

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