Mixed Strategies in Simultaneous and Sequential Play of a 2 Player Game

We take a class of games with two players and two actions which only have mixed strategy Nash equilibria. We show that such games can only have hybrid equilibria if played sequentially with one player moving first. The hybrid equilibrium has the leader p laying a mixed strategy but the follower playing a pure strategy. We apply this result to a game between a debtor and a lender following a loan contract. The debtor can have high or low revenues and has to report his state to the lender. The lender can choose whether or not to undertake a costly audit. With simultaneou s play there is only a mixed strategy Nash equilibrium with random cheating in reports and auditing. With sequential play if the debtor moves first, there is zero auditing and the debtor cheats as much as possible without giving the incentive to audit. We argue that the setting of the game and the valuable first mover advantage of the debtor mean that we should expect the game to be played sequentially with this hybrid outcome. This is important in the context of the loan contract since the hybrid outcome makes the contract renegotiation proof. Alternatively if the timing allowed the lender to move first, then the equilibrium would have the debtor reporting truthfully and the monitor auditing just sufficiently to ensure truthtelling by the debtor. This ha s strong links to the optimal debt contract with no commitment (Mookherjee-Png, 1989 and Jost, 1996). However we argue that the natural timing of events makes the debtor the leader. We then consider other examples and show that the same outcome emerges in matching pennies and in a generic inspection game involving adverse selection in labour markets.Mixed Strategies; Loan Contracts; First Mover Advantage

Looking for new gluon physics at the Tevatron

The impact of nonrenormalizable gluon operators upon inclusive jet cross sections is studied. Such operators could arise in an effective strong interaction Lagrangian from gluon substructure and would induce observable cross section deviations from pure QCD at high transverse jet energies. Comparison of the theoretical predictions with recent CDF data yields a lower limit on the gluon compositeness scale $\Lambda$. We find \Lambda > 2.03 \TeV at $95\%$~CL

Housing Debt and Consumption

The interaction between housing wealth, the financial portfolio of the consumer and consumption is a live issue. Life cycle models with closed form solutions under uncertainty are hard to find. In this paper we find analytical solutions for the effects of house price uncertainty and employment risk on consumption, savings and mortgage finance in a finite horizon life-cycle model. In each period the consumer decides whether to withdraw equity from the house or not, subject to a transaction cost and a constraint on the maximum mortgage loan to house value ratio. Despite risk aversion we findthat, if borrowing is allowed in the financial asset, the prime portfolio effect is the spread between the interest rate and the mortgage rate. House price uncertainty has an ambiguous effect on consumption, which depends on the interest rate differential and house price expectations since future house prices affect future remortgage possibilities. If unsecured debt is not possible, we find that the possibility of future liquidity constraints can reduce mortgage borrowing below the maximum possible.precautionary savings, employment risk, mortgages, housing

Comparative Statics with Consumption Externalities

We consider the comparative statics of consumer demand when there are consumption externalities in one commodity between two individuals. We show that the externality can switch goods which would naturally be normal into inferior goods and as a result th e externality can also lead to Giffen goods. In addition the externality can transform complementarity relations between goods. Thus substitutes can become complements or vice versa once the feedback effects of the externality are taken into account. Next we consider the effect of externalities on Slutsky symmetry and negativity restrictions With consumption externalities there are generalised forms of such restrictions. We derive these both for the two individual case and for cases in which either there are two individuals but all goods may cause externalities or there is a single externality good but H individuals. We relate the generalised symmetry restrictions to the rank conditions of Browning and Chiappori. Finally we consider the effects of consumption externalities on consumer surplus analysis.

Infectious Disease Control by Vaccines Giving Full or Partial Immunity

We use a simple Lotka-Volterra model of the disease transmission process to analyse the dynamic population structure in two scenarios. Firstly a vaccine is available\ on the market at a constant price through time. Secondly, the vaccine is publicly provided. The vaccine works either by giving partial or full immunity to the disease. We analyse market provision for vaccines providing partial immunity and public provision of both types of vaccine. In the case of market provision we find that there may be multiple stationary states and instability. This is in contrast with earlier results under full immunity. In the publicly provided scenario we find that in the partial immunity case a procyclical policy is desirable but for the full immunity case a countercyclical policy is preferable. This is robust to alternative specifications of the basic Lotka-Volterra system.

Truth-telling and the Role of Limited Liability in Costly State Verification Loan Contracts

Recent literature has considered the form of loan contract between two or more risk neutral parties where the revelation principle is inappropriate due to the lack of commitment to an auditing policy by the lender. The privately informed debtor has a stochastic return; once he knows the state realisation, auditing and cheating are determined as Nash equilibria. The literature assumes that this leads to randomised cheating and auditing. In this paper we verify that the contract may involve this randomisation; but that it may also involve truthtelling with random auditing and one or more investors in line with Persons (1996); or a single state independent repayment with no auditing. We define conditions on the state observation cost and the distribution of returns which determine which of these three forms of contract is optimal. We find that under unlimited liability when the loan size is fixed the two investor truthtelling contract dominates all the other forms; and that this is also true when the loan size is optimally chosen. On the other hand under limited liability if the cost of observation is large relative to the lowest state revenue, the random auditing contract or a constrained two investor truthtelling contract may be optimal. The limited liability condition in the constrained truthtelling contracts forces the level of finance to be higher than under unlimited liability.loan contracts, costly state verification, commitment, limited liability

Housing Debt, Employment Risk and Consumption

We consider the interaction between the risk of unemployment, random house prices, consumption and savings. A critical decision is that of refinancing house purchase, up to 100% mortgages are possible. There is also a fixed transaction cost of refinancing. In a CARA framework we derive the value function for a finite horizon, the policy of refinance and the consumption function. Either there is a maximum mortgage or a zero mortgage depending on interest rates, house prices and the transaction cost. The consumption function is linear in wealth and in the uncertainty caused by employment status and house prices of the future. Since there is either 100% or 0% equity withdrawal, consumption jumps when there is refinancing.Precautionary savings; employment risk; mortgages; housing

Conditional and Unconditional Multiple Equilibria with Strategic Complementarities

We take a general model of externalities matching the Cooper & John framework with identical agents. If each agent's payoff depends on a parameter interpreted as the favourableness of the environment, we explore how the number of Nash equilibria varies with this parameter, especially in the cases in which the reaction curves are either concave or convex. In many examples the environmental conditions are themselves endogenous because either market or regulatory forces interact with agents' Nash equilibrium actions. This gives the idea of a simultaneous equilibrium in the environment and players' symmetric actions. We analyse how this generalised equilibrium behaves as a function of some additional parameters conditioning the environmental response to players actions. We show that generally there is a fold bifurcation in these equilibria. We illustrate the principles with two examples from industrial economics (cost spillovers between firms and demand spillovers under imperfect competition).cost spillovers, Nash and Market equilibrium, coordination failure.

Sorting the Good Guys from Bad: On the Optimality of Deterministic Audit with Ex-Ante Information Acquisition

In a costly state verification model under commitment the principal may acquire a costly public and imperfectly revealing signal before or after contracting. If the project remains profitable after all signal realisations, optimally the signal is collected, if at all, after contracting, and it may be acquired randomly or deterministically. Moreover, audit is deterministic and targeted on some signal-state combinations. The paper provides a detailed characterisation of the optimal contract and performs a comparative static analysis of signal acquisition strategy and payoffs with respect to enforcement costs and informativeness of the signal. We explore robustness of the results, including commitment issues.contracts, auditing, ex-ante information