The standard Erdős-Renyi model of random graphs begins with n isolated vertices, and at each round a random edge is added. Parametrizing n 2 rounds as one time unit, a phase transition occurs at time t = 1 when a giant component (one of size constant times n) first appears. Under the influence of statistical mechanics, the investigation of related phase transitions has become an important topic in random graph theory. We define a broad class of graph evolutions in which at each round one chooses one of two random edges {v1, v2}, {v3, v4} to add to the graph. The selection is made by examining the sizes of the components of the four vertices. We consider the susceptibility S(t) at time t, being the expected component size of a uniformly chosen vertex. The expected change in S(t) is found which produces in the limit a differential equation for S(t). There is a critical time tc so that S(t) → ∞ as t approaches tc from below. We show that the discrete random process asymptotically follows the differential equation for all subcritical t < tc. Employing classic results of Crámer on branching processes we show that the component sizes of the graph in the subcritical regime have an exponential tail. In particular, the largest component is only logarithmic in size. In the supercritical regime t> tc we show the existence of a giant component, so that t = tc may be fairly considered a phase transition. Computer aided solutions to the possible differential equations for susceptibility allow us to establish lower and upper bounds on the extent to which we can either delay or accelerate the birth of the giant component. 1 The Achlioptas Problem and Process 1.1

Abstract. This paper develops a theory of revealed preferences over one’s own and others’monetary payo¤s. We introduce “more altruistic than”(MAT), a partial ordering over preferences, and interpret it with known parametric models. We also introduce and illustrate “more generous than ” (MGT), a partial ordering over opportunity sets. Several recent discussions of altruism focus on two player extensive form games of complete information in which the …rst mover (FM) chooses a more or less generous opportunity set for the second mover (SM). Here reciprocity can be formalized as the assertion that an MGT choice by the FM will elicit MAT preferences in the SM and, furthermore, that the e¤ect on preferences is stronger for acts of commision than acts of ommision by FM. We state and prove propositions on the observable consequences of these assertions. Then we test those propositions using existing data from investment games with dictator controls and Stackelberg games and new data from Stackelberg mini-games. The test results provide support for the theory of revealed altruism.

Contents 1. Real Numbers 3 2 3 1.2. Reals, Equality of Reals 5 1.3. The Archimedian Axiom 6 1.4. Nonnegative and Positive Reals 6 1.5. Arithmetical Functions 7 1.6. Comparison of Reals 8 1.7. Non-Countability 10 1.8. Cleaning of Reals 11 2. Sequences and Series of Real Numbers 11 2.1. Completeness 11 2.2. Limits and Inequalities 13 2.3. Series 13 2.4. Redundant Dyadic Representation of Reals 14 2.5. Convergence Tests 15 2.6. Reordering Theorem 17 2.7. The Exponential Series 18 3. The Exponential Function for Complex Numbers 21 4. Continuous Functions 23 4.1. Suprema and In ma 24 4.2. Continuous Functions 25 4.3. Application of a Continuous Function to a Real 27 4.4. Continuous Functions and Limits 28 4.5. Composition of Continuous Functions 28 4.6. Properties of Continuous Functions 29 4.7. Intermediate Value Theorem 30 4.8. Continuity of Functions with More Than One Variable 32 5. Dierentiation 33 5.1. Derivatives 33 5.2. Bounds on the Slope 33 5.3. Properties of Derivatives 34 5

Preemption games are widely used to model patent races, innovation adoption and market entry problems. A previously neglected feature of these problems is that the agents ’ states (e.g. R&D …rms ’ technological improvements) are kept secret and stochastically change over time. We fully characterize equilibrium in preemption games where private information evolves according to Poisson processes, and provide a strategic rationale for the common wisdom that ‘big things happen fast. ’ In the context of patent races we surprisingly …nd that strengthening patent rights need not increase innovation disclosure. Furthermore, we clarify a basic welfare tradeo ¤ between duplication costs and preemption: the former likely take place in early stages of the race, and preemption in later stages.

We show that any symmetric positive denite matrix can be symmetrically scaled by a positive diagonal matrix, or by a diagonal matrix with arbitrary signs, to have arbitrary positive rows sums. The scaling can be constructed by solving an ordinary dierential equation. Key words: positive denite matrices, matrix scaling, diagonal preconditioning, homotopy. PACS: 65F35 7 September 2001 1

This paper develops a common agency model to analyze the strategic interaction between governments in regulating polluting multinationals. We show that when a firm has private information about its production technology relating output to pollution that is difficult to monitor, the information rent extraction behavior of non-cooperative governments will work against the "pollution haven" hypothesis in a Nash equilibrium with or without pooling. The "pollution haven" result is more likely to be reversed in a separating equilibrium than in a pooling equilibrium as a firm's output is further away from the social optimum. This result provides an explanation for why many empirical studies do not support the "pollution haven" hypothesis even after controlling for private non-environmental cost differentials.

This paper develops a common agency model to analyze the strategic interaction between governments in regulating “dirty” multinational firms. These firms possess private information about the degree of pollution associated with their production technologies. The study shows that the strategic behavior between non-cooperative governments, as a result of asymmetric information, works against the “pollution haven” hypothesis. The paper highlights the importance of factors that can dominate environmental costs in a government’s welfare maximization decision rather than those in a firm’s profit maximization decision. The paper also draw implications on the empirical studies of the “pollution haven” hypothesis.