A dynamic programming approach to evolutionarily stable strategy theory

1980 ◽  
Vol 12 (1) ◽  
pp. 3-5 ◽  
Author(s):  
C. Cannings ◽  
D. Gardiner
Keyword(s):  
Dynamic Programming ◽  
Density Function ◽  
Evolutionarily Stable Strategy ◽  
Programming Approach ◽  
Stable Strategy ◽  
War Of Attrition ◽  
Dynamic Programming Approach ◽  
The Individual ◽  
Image Position ◽  
Evolutionarily Stable

In the war of attrition (wa), introduced by Maynard Smith (1974), two contestants play values from [0, ∞), the individual playing the longer value winning a fixed prize V, and both incurring a loss equal to the lesser of the two values. Thus the payoff, E(x, y) to an animal playing x against one playing y, is A more general form (Bishop and Cannings (1978)) has and it was demonstrated that with and there exists a unique evolutionarily stable strategy (ess), which is to choose a random value from a specified density function on [0, ∞). Results were also obtained for strategy spaces [0, s] and [0, s).

A dynamic programming approach to evolutionarily stable strategy theory

1980 ◽  
Vol 12 (01) ◽  
pp. 3-5
Author(s):  
C. Cannings ◽  
D. Gardiner
Keyword(s):  
Dynamic Programming ◽  
Density Function ◽  
Evolutionarily Stable Strategy ◽  
Programming Approach ◽  
Stable Strategy ◽  
War Of Attrition ◽  
Dynamic Programming Approach ◽  
The Individual ◽  
Image Position ◽  
Evolutionarily Stable

In the war of attrition (wa), introduced by Maynard Smith (1974), two contestants play values from [0, ∞), the individual playing the longer value winning a fixed prize V, and both incurring a loss equal to the lesser of the two values. Thus the payoff, E(x, y) to an animal playing x against one playing y, is A more general form (Bishop and Cannings (1978)) has and it was demonstrated that with and there exists a unique evolutionarily stable strategy (ess), which is to choose a random value from a specified density function on [0, ∞). Results were also obtained for strategy spaces [0, s] and [0, s).

The behaviour of strategy-frequencies in Parker's model

1980 ◽  
Vol 12 (01) ◽  
pp. 5-7
Author(s):  
D. Gardiner
Keyword(s):  
Finite Number ◽  
Evolutionarily Stable Strategy ◽  
Pure Strategy ◽  
Payoff Matrix ◽  
Stable Strategy ◽  
Expected Payoff ◽  
Pure Strategies ◽  
Image Position ◽  
Evolutionarily Stable

Parker's model (or the Scotch Auction) for a contest between two competitors has been studied by Rose (1978). He considers a form of the model in which every pure strategy is playable, and shows that there is no evolutionarily stable strategy (ess). In this paper, in order to discover more about the behaviour of strategies under the model, we shall assume that there are only a finite number of playable pure strategies I 1, I 2, ···, I n where I j is the strategy ‘play value m j ′ and m 1 < m 2 < ··· < m n . The payoff matrix A for the contest is then given by where V is the reward for winning the contest, C is a constant added to ensure that each entry in A is non-negative (see Bishop and Cannings (1978)), and E[I i , I j ] is the expected payoff for playing I i against I j . We also assume that A is regular (Taylor and Jonker (1978)) i.e. that all its rows are independent.

On the dynamic programming approach to Pontriagin's maximum principle

1968 ◽  
Vol 5 (3) ◽  
pp. 679-692 ◽  
Author(s):  
Richard Morton
Keyword(s):  
Dynamic Programming ◽  
Maximum Principle ◽  
The State ◽  
Dynamic Programming Principle ◽  
Programming Approach ◽  
Bellman Function ◽  
State Variables ◽  
Dynamic Programming Approach ◽  
Minimum Value ◽  
Image Position

Suppose that the state variables x = (x1,…,xn)′ where the dot refers to derivatives with respect to time t, and u ∊ U is a vector of controls. The object is to transfer x to x1 by choosing the controls so that the functional takes on its minimum value J(x) called the Bellman function (although we shall define it in a different way). The Dynamic Programming Principle leads to the maximisation with respect to u of and equality is obtained upon maximisation.

The allocation of resources in a multiple-trial war of attrition conflict

1996 ◽  
Vol 28 (3) ◽  
pp. 933-964 ◽  
Author(s):  
J. C. Whittaker
Keyword(s):  
Evolutionarily Stable Strategy ◽  
Pure Strategy ◽  
Allocation Of Resources ◽  
Stable Strategy ◽  
Fixed Amount ◽  
War Of Attrition ◽  
Evolutionarily Stable

We consider a model in which players must divide a fixed amount of resource between a number of trials of an underlying contest, with the underlying contest based on the War of Attrition. We are able to find the unique ES set (a simple generalisation of the idea of an evolutionarily stable strategy) in certain circumstances: in particular we find the conditions under which the ES set may contain a pure strategy.

On the dynamic programming approach to Pontriagin's maximum principle

1968 ◽  
Vol 5 (03) ◽  
pp. 679-692
Author(s):  
Richard Morton
Keyword(s):  
Dynamic Programming ◽  
Maximum Principle ◽  
The State ◽  
Dynamic Programming Principle ◽  
Programming Approach ◽  
Bellman Function ◽  
State Variables ◽  
Dynamic Programming Approach ◽  
Minimum Value ◽  
Image Position

Suppose that the state variables x = (x 1,…,x n )′ where the dot refers to derivatives with respect to time t, and u ∊ U is a vector of controls. The object is to transfer x to x 1 by choosing the controls so that the functional takes on its minimum value J(x) called the Bellman function (although we shall define it in a different way). The Dynamic Programming Principle leads to the maximisation with respect to u of and equality is obtained upon maximisation.

The behaviour of strategy-frequencies in Parker's model

1980 ◽  
Vol 12 (1) ◽  
pp. 5-7 ◽  
Author(s):  
D. Gardiner
Keyword(s):  
Finite Number ◽  
Evolutionarily Stable Strategy ◽  
Pure Strategy ◽  
Payoff Matrix ◽  
Stable Strategy ◽  
Expected Payoff ◽  
Pure Strategies ◽  
Image Position ◽  
Evolutionarily Stable

Parker's model (or the Scotch Auction) for a contest between two competitors has been studied by Rose (1978). He considers a form of the model in which every pure strategy is playable, and shows that there is no evolutionarily stable strategy (ess). In this paper, in order to discover more about the behaviour of strategies under the model, we shall assume that there are only a finite number of playable pure strategies I1, I2, ···, In where Ij is the strategy ‘play value mj′ and m1 < m2 < ··· < mn. The payoff matrix A for the contest is then given by where V is the reward for winning the contest, C is a constant added to ensure that each entry in A is non-negative (see Bishop and Cannings (1978)), and E[Ii, Ij] is the expected payoff for playing Ii against Ij. We also assume that A is regular (Taylor and Jonker (1978)) i.e. that all its rows are independent.

The allocation of resources in a multiple-trial war of attrition conflict

1996 ◽  
Vol 28 (03) ◽  
pp. 933-964 ◽  
Author(s):  
J. C. Whittaker
Keyword(s):  
Evolutionarily Stable Strategy ◽  
Pure Strategy ◽  
Allocation Of Resources ◽  
Stable Strategy ◽  
Fixed Amount ◽  
War Of Attrition ◽  
Evolutionarily Stable

We consider a model in which players must divide a fixed amount of resource between a number of trials of an underlying contest, with the underlying contest based on the War of Attrition. We are able to find the unique ES set (a simple generalisation of the idea of an evolutionarily stable strategy) in certain circumstances: in particular we find the conditions under which the ES set may contain a pure strategy.

Sign in / Sign up

Export Citation Format

Share Document