coordination and carbon prices

Some applied game theorists write in Nature that an international agreement to an effective price of carbon emissions would be more likely to work out than an attempt at an international cap-and-trade scheme, and their arguments seem basically right to me.  In at least some sense, though, I think we could do better.

Suppose, since this seems to be the discussion being had at the moment, that we can treat each country as a rational agent, and suppose each country knows the effective price in each other country. Let w_i denote more or less the size of a country normalized so that ∑_i w_i = 1 and suppose each country wants to maximize U_i = aP-bP²-p_i where P=∑ w_ip_i ; this at least captures the idea that each country would like to minimize its own price but (in the relevant range) wants the world price to be higher, but subject to diminishing returns. If we each agree to set our price at c+dP, where 0<d<1, then if country i tries to cheat by δ, that reduces P and thus reduces other countries' prices as well, resulting in an overall decrease of δ/(1-d); its own cost is aw_iδ/(1-d)-2bPw_iδ/(1-d)-δ, which is positive if (a-2bP)w_i>(1-d). If you're trying to optimize ∑w_i U_i and you set d=1-w_i, then country i will want to comply with the optimal price as long as everyone else does.  Different countries will have different w_i, and for reasons made clear in the article you really couldn't give different countries different values of d and expect it to work out well — you'd have the same problems currently encountered with the cap-and-trade negotiations.  1/3, however, is a reasonable maximum value, which suggests that setting d to at least 2/3 would substantially reduce the incentive to cheat.

I suspect two potential problems with my scheme, especially vis-a-vis theirs: 1) mine is a bit more complicated, which they note often empirically impedes agreement and coordination, and 2) mine may be more susceptible to problems with monitoring; countries would have a stronger incentive in my scheme than in theirs (depending on what enforcement mechanisms they envision) to make the rest of the world think their price is higher than it really is.