A Pairwise Averaging Procedure with Application to Consensus Formation in the Deffuant Model

Place a water glass at each integer point, the one at the origin being full and all others empty, and consider averaging procedures where we repeatedly pick a pair of adjacent glasses and pool their contents, leaving the two glasses with equal amounts but with the total amount unchanged. Some simple results are derived for what kinds of configurations of water levels are obtainable via such procedures. These are applied in the analysis of the so-called Deffuant model for social interaction, where individuals have opinions represented by numbers between 0 and 1, and whenever two individuals interact they take a step towards equalizing their opinion, unless their opinions differ beyond a fixed amount theta in which case they make no adjustment. In particular, we reprove and sharpen the recent result of Lanchier which identifies the critical value theta(c) for consensus formation in the Deffuant model on on Z to be 1/2.