Dividends and weighted values in games with externalities

We consider cooperative environments with externalities (games in partition function form) and provide a recursive definition of dividends for each coalition and any partition of the players it belongs to. We show that with this definition and equal sharing of these dividends the averaged sum of dividends for each player, over all the coalitions that contain the player, coincides with the corresponding average value of the player. We then construct weighted Shapley values by departing from equal division of dividends and finally, for each such value, provide a bidding mechanism implementing it. ∗We thank Guillaume haeringer for his comments. We gratefully acknowledge the financial support from SEJ2006-00538-ECON. Macho-Stadler and Pérez-Castrillo also acknowledge the finantial support from 2005SGR-00836, Consolider-Ingenio CSD2006-00016 and Barcelona Economics-Xarxa CREA. This paper is dedicated to Michael Mashler, a small proof of our appreciation to a great scholar and a wonderful person. †Dep. of Economics & CODE, Universitat Autònoma de Barcelona. Email: [email protected] ‡Corresponding author. Dep. of Economics & CODE, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain. Email: [email protected]. Ph.: (34) 93 581 1405 and (34) 93 581 2461. Fax: (34) 93 581 2461. §Dep. of Economics, Ben-Gurion University of the Negev. Email: [email protected]