Game Theory: Contention and Cross-Effects (Part 3)

Static Game, Prisoners Dilemma and Strategy Profile

Introduction

In order to maximize your well being, you not only think of your actions but also guess what other players are doing, in order to maximize your reward. Your contenders are also no less rational than you, they take decisions in a similar way.
Let’s introduce some theoretical framework to formalize this new decision problems.

Static Game

We need some assumptions so that each player in strategic environment can behave rationally. We assume that all player in strategic game know:
1) All the possible actions of all the players
2) All the possible outcomes
3) Cross-effects
4) The preferences of each and every player over outcomes

We can introduce a decision problem where players ( More than one) have to choose actions from action space and the combinations of those such choices results in outcomes. Each player in the decision problem have preferences for these outcome. We have set of actions for each players and set of payoff functions for each player. Payoff functions in normal-form game: which gives the payoff value for combination of actions chosen by each player in the normal game.
Pure strategy is simply plan of actions. Let’s discuss simple example to make things more clearer for the readers.

Example: Prisoners Dilemma

Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent. The possible outcomes are:

  1. If A and B each “betray”(BE) the other, each of them serves two years in prison
  2. If A betrays B, but B “remains silent”(RS), A will be set free and B will serve three years in prison (and vice versa)
  3. If A and B both remain silent, both of them will serve only one year in prison (on the lesser charge).

Players: N= {A,B}
Strategic sets: S= {BE, RS}
Payoffs: vA(sA,sB) be the payoff of player A and vB(sA,sB) be the payoff of player B

vA(BE,BE) = vB(BE,BE) = -2
vA(RS,RS) = vB(RS,RS) = -1
vA(BE,RS) = vB(RS,BE) = 0
vA(RS,BE) = vB(BE,RS) = -3
Its more convenient to represent these numbers in matrix representations.

Rows: Represent players A strategies
Columns: Represents player B strategies
Matrix Entries: payoff of A/B

Like the decision tree structure introduced in the last blog, this matrix representation completely defines a strategic game.

rock-paper-scissors.

Players: N = {1, 2}
Strategic sets: S= {R, P, S}
payoff matrix:

Strategy Profile: It is basically set of actions taken by player, there are 9 possible strategy profiles. For example {R,R} is one strategy profile, which implies Player 1 and 2 both decides to choose rock.

vᵢ(s) is payoff of a player i from a profile of strategies s = (s₁, s₂, . . . , sᵢ₋₁ , si, sᵢ₊₁, . . , sn). We define strategy profile s₋ᵢ ∈ S₋ᵢ as a particular possible profile of strategies for all players who are not i.

Conclusion

Thanks for you time.

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Kowshik chilamkurthy

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RL | ML | ALGO TRADING | TRANSPORTATION | GAME THEORY

Nerd For Tech

NFT is an Educational Media House. Our mission is to bring the invaluable knowledge and experiences of experts from all over the world to the novice. To know more about us, visit https://www.nerdfortech.org/.