# Game Theory: Decisions, Interaction and Evolution

## James N. Webb

Language: English

Pages: 0

ISBN: B002RM2WXC

Format: PDF / Kindle (mobi) / ePub

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U C D L 1, 3 4, 0 2, 5 P2 M 4, 2 0, 3 3, 4 R 2, 2 4, 1 5, 6 From the start, neither player has any dominated strategies leading to the 4.4 Nash Equilibria 69 maximally imprecise prediction that anything can happen. (It is in this sense that the solution is “trivial”.) Nevertheless, there is an “obvious” solution to this game, namely (D, R), which maximises the payoﬀ to both players. Is it possible to define a solution in terms of something other than the (iterated) elimination of

The set of alternative actions available will be denoted A. This will either be a discrete set, e.g., {a1 , a2 , a3 , . . .}, or a continuous set, e.g., the unit interval [0, 1]. Definition 1.5 A payoﬀ is a function π: A → R that associates a numerical value with every action a ∈ A. 6 1. Simple Decision Models Definition 1.6 An action a∗ is an optimal action if π(a∗ ) ≥ π(a) ∀a ∈ A (1.1) a∗ ∈ argmax π(a) . (1.2) or, equivalently, a∈A That is, the optimal decision is to choose an a∗ ∈ A

Naturalis Principia Mathematica: “We are to admit no more causes of natural things than such as are both true and suﬃcient to explain their appearances”. 8.4 Pairwise Contest Games 149 a polymorphic population a fraction x of the population use σH = (1, 0) and a fraction 1 − x use σD = (0, 1). We will consider only monomorphic populations for the moment. At various times, individuals in this population may come into conflict over a resource of value v. This could be food, a breeding site,

has an equal chance of being an owner or an intruder. (In genetic terms, the genes that are currently in an owner may find themselves passed on to an oﬀspring that has yet to find a resource to control.) With these assumptions, we can derive the payoﬀ table shown in Figure 8.1. For example, consider the expected payoﬀ to players using HH against opponents who use HD. Half the time they will be the owner using H against an intruder who uses D, and half the time they will be an intruder using H

investor is indiﬀerent between a1 and a2 (i.e., both a1 and a2 are optimal). 1.3 Modelling Rational Behaviour 11 Exercise 1.8 Each day a power company produces u units of power at a cost of c dollars per unit and sells them at a price p dollars per unit. Suppose that demand for power is exponentially distributed with mean d units, i.e., x 1 . f (x) = exp − d d If demand exceeds supply, then the company makes up the shortfall by buying units from another company at a cost of k dollars per unit