A New Sufficient Condition for Uniqueness in Continuous Games

Consider the class of games in which each player chooses a strategy from a connected subset of the real line. Many oligopoly models fall into this class. In many of these applications, it would be useful to show that an equilibrium was unique, or at least to have a set of conditions under which uniqueness would hold. In this paper, we first prove a uniqueness theorem that is slightly less restrictive than the contraction mapping theorem for mappings from the subsets of the real line onto itself, and then show how uniqueness in the general game can be shown by proving uniqueness using an iterative sequence of R-to-R mappings. This iterative approach works by considering the equilibrium for an m-player game holding the strategies of all other players fixed, starting with a two-player game. If one can show that the m-player game has a unique equilibrium for all possible values for the remaining players strategies, then one can add one player at a time and consider the R-to-R mapping from that player’s strategy on to the unique equilibrium of the first m players and back onto the (m+1)th player’s strategy. We then show how a general condition for each one of this sequence of mappings to have a unique equilibrium is that the leading principal minors of a matrix derived from the Jacobean matrix of best-response functions be positive, and how this general condition encompasses and generalises some existing uniqueness theorems for particular gamesUniqueness; Continuous Games; Oligopoly

A New Existence and Uniqueness Theorem for Continuous Games

This paper derives a general sufficient condition for existence and uniqueness in continuous games using a variant of the contraction mapping theorem applied to mapping from a subset of the real line on to itself. We first prove this contraction mapping variant, and then show how the existence of a unique equilibrium in the general game can be shown by proving the existence of a unique equilibrium in an iterative sequence of games involving such R-to-R mappings. Finally, we show how a general condition for this to occur is that a matrix derived from the Jacobean matrix of best-response functions be have positive leading principal minors, and how this condition generalises some existing uniqueness theorems for particular games.Existence; Uniqueness; Continuous Games; Contraction Mapping Theorem

A Note on the Probability of Winning a Lottery when the Number of Competitors is a Binomial Random Variable

Consider a model in which a consumer faces a lottery with j other people for a prize, so that the probability of winning the prize is 1/(j+1). Now let j be a random variable, determined by the binomial distribution. Specifically, let there be n potential competitors for the consumer in the lottery, each with an independent probability of ? of being a competitor. In this note, we show how the resulting expression for the expected value of 1/(j+1) using binomial probabilities can be simplified by means of the binomial theorem.Binomial Distribution; Binomial Theorem; Lottery

A Method for Inferring Batting Conditions in ODI Cricket from Historical Data

This paper is part of a wider research programme using a dynamic-programming approach to modelling the choices about the amount of risk to take by batting and bowling teams in One Day International cricket. An important confounding variable in this analysis is the ground conditions (size of ground, nature of pitch and weather conditions) that affect how many runs can be scored for a given amount of risk. This variable does not exist in our historical data set and would regardless be very difficult to accurately observe on the day of a match. In this paper, we consider a way of estimating a distribution for the ground conditions using only the information contained in the first-innings score and the result of the match. The approach uses this information to estimate the importance of ground conditions in the determination of first innings total scores. We assume a functional form for a model of first innings scores and we estimate the parameters of our model using Monte Carlo methods. We test the impact of a significant rule change and we apply our findings to selected matches before and after the new rules came into play.ODI Cricket; Batting Conditions; Nuisance Variable

A Note on Log Concave Survivor Functions in Auctions

In a standard English auction in which bidders’ valuations are independently drawn from a common distribution, a standard regularity condition is that the survivor function of the distribution be log concave. In an auction where the seller sets a fixed price, the equivalent condition requires log concavity of a survivor function derived from the primitive distribution. In this note we show that log concavity of the primitive survivor function implies log concavity in the derived functions. This result is of interest when studying on-line auctions that combined aspects of fixed-price and English auctions.English Auction; Log Concavity; Survivor Function

A Critique of Wolak’s Evaluation of the NZ Electricity Market: The Incentive to Exercise Market Power with Elastic Demand and Transmission Loss

This paper is the second in a symposium of papers that examine the 2009 report by Frank Wolak into the New Zealand electricity market. In this paper, we discuss the Report’s measures of the ability and incentives of generators to exercise unilateral market power. We show that the construction and interpretation of these measures are highly sensitive to some key assumptions, particularly those concerning the elasticity of demand for electricity in the wholesale market and the amount of transmission loss on the national grid.Wolak Report; electricity markets; market power

A Critique of Wolak’s Evaluation of the NZ Electricity Market: Introduction and Overview

This paper is the first in a symposium of papers that examine the 2009 report by Frank Wolak into the New Zealand electricity market. The Wolak report concluded that there had been a cumulative total of $4.3b (NZD) of overcharging in the New Zealand wholesale market over a period of seven years. In this paper, we introduce the Wolak findings in the context of the salient features of the New Zealand market, and explain that this headline figure is highly sensitive to some (quite unrealistic) assumptions about the structure of this market. The papers that follow this introduction (Hogan and Jackson, 2011, and Evans and Guthrie, 2011) examine Wolak’s methodology and its empirical application.Wolak Report; electricity markets; market power

Vertical Integration and Market Power in Electricity Markets

Vertical separation of generation from electricity retailing has often been required as a condition of electricity market liberalisation. A well-developed and liquid contracts market is similarly suggested as necessary to manage the resulting wholesale market risks which risks are further exacerbated by competition. Such contracts markets are rare however and increasingly evidence is emerging that vertical integration is associated not just with improved wholesale market risk management but also reduced wholesale market power. This paper develops a theoretical model showing that non-vertically integrated generators will over-report their inverse supply curves with the incentive to over-report increasing with the firm's share of generating capacity. Conversely in a vertically integrated industry no over-reporting occurs when integrated firms have balanced shares in wholesale and retail markets. In general firms whose share of generating capacity is higher (lower) than their retail market share will over-report (under-report) their inverse supply functions. Integration is found to affect retail electricity prices only via its effect on retail marginal costs. We find that retail prices are higher with vertical separation than with either balanced integration or full integration without a wholesale market. These results suggest a re-evaluation of the importance of generator wholesale market power in vertically integrated electricity industries and of measures to improve retail market competitiveness under either vertical integration or separation

The Fixed Price Offer Mechanism in Trade Me Online Auctions

The Fixed Price Offer (FPO) mechanism in Trade Me auctions allows sellers to make a take-it-or-leave-it offer at the conclusion of an unsuccessful auction. We investigate the effects of the FPO option on strategies and outcomes in independent-value auctions. The FPO option induces some bidders with a value above the seller’s reserve to wait for an FPO instead of bidding. Overall, the FPO option increases the probability of sale but reduces expected seller revenue compared to a standard auction. The impact of the FPO option is reduced when the number of bidders increases.fixed price offer; private value auction; on-line auction; optimal reserve price; second chance offer

Downward wage rigidity

There has recently been considerable discussion about the ability of inflation to facilitate the adjustment of prices and wages and thus enhance economic performance. The discussion centres on whether wages are downwardly rigid. Wages are said to be downwardly rigid if it is difficult for the wages of some workers to fall despite underlying supply and demand pressures for decreases. Some authors have suggested that if downward nominal wage rigidity is prevalent it would be desirable to select a positive rate of inflation as the target for monetary policy. In this article, the authors evaluate the wage-rigidity hypothesis. They first examine the empirical evidence to assess whether the degree of downward rigidity is significant in Canada. They then analyze some key assumptions of the wage-rigidity hypothesis and its implications for employment. They also look at the empirical evidence on whether the combination of downward wage rigidity and low inflation has reduced employment.