【正文】 basis for proportional cutbacks to improve CPR performance. In particular, we establish the existence of a range of proportional cutbacks to move users from an inefficient game equilibrium to an improved oute — in some cases, even an optimum. The basic criterion is that the proportional cutback be not too large, for which we give an explicit bound. Section 4 develops a model for a parametric class of CPR situations to illustrate these results. We apply a specific calibration of the model to study the workings of proportional cutbacks in the Kyoto Protocol. Section 5 extends our analysis of oneshot games to timedependent CPR situations, where users are heterogeneous in benefits and prior appropriation rights to the CPR are present. The Section 6 offers concluding ments. 2. Proportional cutbacks in the field In the late 1970s, satellite photos revealed for the first time the presence of ozone holes in the earth’s atmosphere over the North and South Poles. Atmo spheric scientists immediately began researching probable causes for the ozone holes, as well as their likely implications for life on the pla. Since atmospheric ozone shields life on the pla from lethal radiation from space, the plete disappearance of ozone from the atmosphere would have catastrophic consequences for the biosphere. Even worse, the ozone holes were observed to be expanding — with the ozone hole over the South Pole covering almost the entire continent of Antarctica during the Southern Hemisphere’s winter, and the hole over the North Pole somewhat smaller. By 1985, atmospheric chemists had provided a causal explanation for the ozone holes beyond any reasonable doubt: as propellants in aerosol cans and as coolants in air conditioners and refrigerators were the culprits. Each molecule of CFC, released at the earth’s surface, had the capacity to rise to the ozone layer and chemically unbind over 4000 molecules of ozone. CFCs, which had e to mercial prominence in this century, had in a short time created the ozone holes. In response to the ozone situation, negotiations began on an international treaty to ban the mercial production of CFCs by all countries. In essence, this ban represents the limiting case of a proportional cutback — a 100% cutback. Such cutbacks would certainly increase costs of refrigeration and air conditioning, both for producers and consumers. Moreover, new propellants would have to be developed for the multibilliondollar aerosolcan industry worldwide. The costs of these cutbacks would be concentrated in the developed countries, but they would also be felt in the developing countries. Treaty negotiations moved swiftly, culminating in the Montreal Protocol on Substances that Deplete the Ozone Layer, adopted by over 125 countries in 1987. The Montreal Protocol phases out the production and distribution of CFCs over a 10year period — freon is just now disappearing from refrigerators and air conditioners in the United States _UNEP, 1991.. In the case of the ozone holes, the use of proportional cutbacks must be counted a considerable success. The holes themselves have stopped growing, although it is estimated it will take at least a century before they disappear altogether. The benefits to the quality of life on earth outweigh the losses to CFC producers and consumers. Indeed, the latter have succeeded in developing close substitutes for CFCs without serious atmospheric consequences. Not all international attempts to