A Stable Price Equilibrium

Overview. Signatories of a pricing treaty need to be willing to accept the global price established by the treaty in order for the treaty to be stable. But this means a country that say it will accept only and extremely low price, can remain outside the treaty and free-ride to some extent.

This section shows that a well-structured pricing treaty can change the climate game from a prisoners’s dilemma (PD) (a public goods game) to a game of chicken, aka hawk-dove (HD). Fortunately, the set of non-cooperative strategies, which is the only Nash equilibrium (NE) in a PD is not a NE in the HD game. Instead the, the HD game is likely to have a fairly cooperative mixed-strategy NE.

A global pricing treaty establishes a global price (and more) that the signatories will comply with. Since countries cannot be compelled to sign, the price must be agreeably too all signatories. Hence an obvious mechanism (similar to a provision point mechanism) for getting unanimous agreement on a price is to specify an emissions coverage, say 70%, and implement the highest price that can achieve that coverage with unanimity.

Obviously a tradeoff must be made. If 100% coverage were required, the price would need to be very low, quite possibly zero.

A convenient way to implement such a treaty is to announce, the coverage requirement, C%, (or a formula for C that depends on the pledges) and ask all countries to pledge a price they will comply with if C% of the world also complies. The pledges are then ordered from highest to lowest and accepted in that order until at least C% of emissions have been covered. The last pledge accepted is the lowest accepted and sets the global (treaty) price. In this way all treaty signers will have agreed to accept the price they have signed on for — the “global” price.

Gaming the Price Treaty. If C were set to 100%, there would be no reason to game the price. Every country will consider the question, “I we could set the price, what price would be choose?” Call country i’s answer, P(i). All countries will pledge P(i), unless they know for sure that P(i) will not set the price. But in that case, they will either know that P(i) > P, the ultimate global price, or P(i) < P. I the first case the will want a higher P, but by pledging differently (lower than P), they could only make it lower. So they will choose not to affect the price. A similar argument holds of they want a lower price than P. The only remaining case is when they are country n, the last country accepted and the one that sets the price. In this case since there pledge sets P, and they want P = P(i), the will (honestly) pledge P(i).

Unfortunately, when C < 100%, a new motive enters the picture, and their is reason for countries to pledge a different Pg(i) than P(i).

which means there pledge will honestly reflect their preference. Why? 

T-n in a world with N equal-size (but not identical) countries as one that sets the global price at P(n), the price named by the country with the n^th lowest price.⁽¹⁾Realistically n would be defined to weight countries by their emissions.

The problem is to make the best trade-off between a higher price and greater coverage.

Three ways to avoid a race to the bottom:

1. use the Green Fund
2. punish defectors (those pledging P < P(n).)
3. reduce n, the price cutoff (if the Green Fund becomes too big)

Enforcement: #1 (a carrot) and #2 (a stick) can both be considered enforcement because both can raise the price level at which a country wants to be inside the treaty. The Green Fund is ambiguous on this score because it will lower this threshold price for those who pay into it, but generally these are countries with a higher threshold.

First assume a given enforcement level. This leaves us with the question of how to set n, the coverage cutoff.

[There is a remaining enforcement question: can countries profit by pledging a high price and then reneging? Call this failure to comply.]

Understanding a country’s payoff function.

• Say a country pledges P. One of three things will happen.
  • It will sign the treaty a P* ≤ P and comply.
  • It will sign and renege
  • It will not sign because P ≤ P*.
• But the payoff of these outcomes depends on P* and on compliance.
• Let’s assume there is compliance because:
  • The country is happy with P* as the global price.
  • It’s high pledge would not raise the global price much, or for long, so there is not much to be gained by over-stating and reneging.
• In this case (Case A) only two things can happen
  • It will sign the treaty a P* ≤ P and comply.
  • It will not sign because P ≤ P*.
• First consider a simpler problem:
  • The country must sign and all signers must comply,
  • but it can set P* unilaterally.
  • Call it’s optimal P* under these conditions:  P1*.
  • Should this be it’s pledge in Case A?
• It has two sensible alternatives: pledge P1* (Case A1) or pledge 0 (Case A2).
  • Case A1 results in a higher P*,
  • but more abatement cost from compliance.
• Let us find the condition for choosing to join the treaty
  • Let P(i) for 0<i≤N be the pledge function.
  • Let P’(n) = P(n) − P(n-1)
  • Then we need Benefit (P'(n)) > cost of abate(P(n)) — roughly speaking.

Here is a simpler, similar game, which illustrate the problem with finding an equilibrium.

1. Two players, each can contribute $0 to $10 to “the pot”
   1. The one who contributes more will have her money quadrupled and redistributed equally to both players.
   2. The other one gets their money back from the pot.
   3. If there is a tie, the both get half back and the other half quadrupled.
2. There are several NE, the two canonical ones are (0, 10) and (10, 0).
   1. This creates a coordination problem.
   2. The problem is destructive because, if the each try for the equilibrium they prefer, the get no payoff.
3. So it looks like the non-NE, (0,0), might prevail in a one shot game.
   1. Is this right. Is there a mixed strategy?
   2. In a repeated game can we get more cooperation?
   3. What’s the simplest version of this game?
      1. Players can only choose 0 or 1.
      2. (1,0) and (0,1) are the only NE’s
      3. Is this “Chicken”? with 0 = go straight, 1 = swerve?
         1. Looks like it is.
         2. So there is a mixed strategy
   4. What does the mixed strategy look like in the (0, 1, 2) game?
   5. Or what if we have more players and they mix (0 and P1*).
4. What if players are not completely rational?
   1. They may not want to appear totally uncooperative by bidding zero. This may help.
   2. The may feel they should cooperate some, even if not in the treaty
   3.