Nash Bargaining Over Margin Loans to Kelly Gamblers

I derive practical formulas for optimal arrangements between sophisticated stock market investors (continuous-time Kelly gamblers or, more generally, CRRA investors) and the brokers who lend them cash for leveraged bets on a high Sharpe asset (i.e., the market portfolio). Rather than, say, the broker posting a monopoly price for margin loans, the gambler agrees to use a greater quantity of margin debt than he otherwise would in exchange for an interest rate that is lower than the broker would otherwise post. The gambler thereby attains a higher asymptotic capital growth rate and the broker enjoys a greater rate of intermediation profit than would be obtained under non-cooperation. If the threat point represents a complete breakdown of negotiations (resulting in zero margin loans), then we get an elegant rule of thumb: r L * = 3 / 4 r + 1 / 4 ν − σ 2 / 2 , where r is the broker’s cost of funds, ν is the compound-annual growth rate of the market index, and σ is the annual volatility. We show that, regardless of the particular threat point, the gambler will negotiate to size his bets as if he himself could borrow at the broker’s call rate.

Formal Contracts, Relational Contracts, and the Threat-Point Effect

We investigate whether formal contracts can help in resolving the holdup problem by studying repeated transactions between a seller and a buyer. Contrary to previous findings, we demonstrate that a simple fixed-price contract based on product delivery is of value even when relation-specific investment is purely cooperative. Furthermore, we show that focusing our attention on fixed-price contracts as a form of formal contracts is without loss of generality. The key driving force is a possibility that the relation-specific investment decreases the surplus under no trade. This possibility, although very plausible, has been largely ignored in previous analyses of the holdup problem. (JEL C78, D23, D86)

The role of fixed cost in international environmental negotiations

ABSTRACTWe investigate the relative efficiency of an agreement based on a uniform standard without transfers and one based on differentiated standards with transfers when strictly identical countries deal with transboundary pollution. We especially ask what role fixed cost plays. Two approaches are examined: the Nash bargaining solution, involving two countries, and the coalition formation framework, involving numerous countries and emphasizing self-enforcing agreements. In the former, in terms of welfare, strictly identical countries may wish to reduce their emissions in a non-uniform way under the differentiated agreement. For this result to hold, the fixed cost of investment in abatement technology must be sufficiently high. The nature of the threat point of negotiations, however, also plays a crucial role. As concerns global abatement, the two countries abate more under the uniform agreement than under the differentiated one. In terms of coalition formation when numerous countries are involved, a grand coalition could emerge under a differentiated agreement.

A NOTE ON REPEATED GAMES WITH VANISHING ACTIONS

A two-person general-sum repeated game with vanishing actions is an infinitely repeated game where the players face the following restrictions. Each action must be used by player k ∈ {1,2} at least once in every rk ∈ ℕ consecutive stages, otherwise the action vanishes for the remaining play. We assume that the players wish to maximize their limiting average rewards over the entire time-horizon. A strategy-pair is jointly convergent if for each action pair a number exists to which the relative frequency with which this action pair is chosen, converges with probability one. A pair of feasible rewards is called individually rational if each player receives at least the threat-point reward, i.e., the amount which he can guarantee himself regardless of what the opponent does given r1, r2 and the actions available in the long run. In a repeated game with vanishing actions, there may exist multiple threat points which are endogenous to the play. We prove that all individually-rational jointly-convergent pure-strategy rewards can be supported by an equilibrium. Furthermore, each convex combination of individually-rational jointly-convergent pure-strategy rewards, can be supported by an equilibrium for m × n-games provided r1 > m ≥ 2, r2 > n ≥ 2.