Skip to main content


The counterexample is the following.

Suppose that an endowment of an economy is 2 right hand (RH) gloves and 1 left hand (LH) gove. A pair of gloves is worth $1.00.

The attribute core assigns a RH glove a payoff of zero and a LH glove a payoff off $1; the attribute core is the singleton set {(p(R), p(L))=(0,1)}.

Now let us suppose that there are two people in the economy. One person owns both the two RH gloves and the other owns the LH glove. The Engl-Scotchmer core concept, essentially the same as the well known concept of subsidy free prices, is the set of nonnegative vectors

{(p(R),p(L)): p(R)+p(L)=1}.

These two sets are not the same. The Engl-Scotchmer concept takes people as coalitions, so their coalitions are only the one-person and two person coalitions. The attribute core (or the equal treatment core of the 3 player game where the gloves themselves are players) has more players and a larger number of coalitions.

Of course the concepts are related and also related to subsidy free prices, to price taking equilibrium prices and to equal treatment cores of games. While the equal treatment core of a game is well prior to any research of mine, it has played roles in a number of my papers (alone and with others) since 1979.

In addition, the models are related. Both essentially Shapley-Shubik (1966) models of TU economies. Moreover, both papers demonstrate convergence results for approximate cores, as does Shubik and Wooders (1982) and Bennett and Wooders (1979) and other earlier working papers of mine. My paper uses the concept of the attribute core to obtain convergence -- of a different sort than that of E&S -- of approximate cores to competitive payoffs in TU economics.

And my paper cites theirs! And has since 1991. What is their problem?