1. 1-1 Game Theory Intro - TCP Backoff
  2. 1-2 Self-Interested Agents and Utility Theory
  3. 1-3 Defining Games
  4. 1-4 Examples of Games
  5. 1-5 Nash Equilibrium Intro
  6. 1-6 Strategic Reasoning
  7. 1-7 Best Response and Nash Equilibrium
  8. 1-8 Nash Equilibrium of Example Games
  9. 1-9 Dominant Strategies
  10. 1-10 Pareto Optimality

Game Theory

Week 1: Introduction and Overview

https://www.coursera.org/learn/game-theory-1/home/week/1

1-1 Game Theory Intro - TCP Backoff

从 TCP 协议中的退避机制（backoff mechanisn）引出博弈：

A game in general is any interaction between two or more people where the outcomes of the interaction depend on what everybody does and everybody has different levels of happiness for the different outcomes.

博弈就是两或多人间的互动，其中互动的结果取决于每个人的行为，并且每个人对于不同的结果都会有不同的愉悦度。

1-2 Self-Interested Agents and Utility Theory

utility function is, a mathematical measure that tells you how much the agent likes or does not like a given situation.

收益函数是指：一个数学的评估方法，用于决定一个代理对于特定情况的喜恶程度。

And the decision theoretic approach which is what underlies modern game theory, says that you're going to try to act in the way that maximizes your expected or average utility.

对于这种喜好的理论决定方法，就是现代博弈论的基础。这个基础就是，每个人都试图将期望效益最大化。

1-3 Defining Games

  • Normal Form (a.k.a. Matrix Form, Strategic Form)

    Finite, n-person normal form game: \(<N, A, u>\)

    • Players

      \(N = \{1, \cdots, n\}\)

    • Actions

      Action set for player \(i\): \(A_i\)

    • Payoffs

      Utility function or Payoff function for player \(i\): \(u_i: A \mapsto \mathbb{R}\)

  • Extensive Form

    • Timing

    • Information

1-4 Examples of Games

  • Games of Cooperation

  • Coordination Game

  • General Games

1-5 Nash Equilibrium Intro

  • Keynes Beauty Contest Game

  • Keynes Beauty Contest Game: The Stylized Version

1-6 Strategic Reasoning

Nash Equilibrium:

纳什均衡：

  • A consistent list of actions

  • Each player's action maximizes his or her payoff given the actions of the others

    给定其他人的行为，每个玩家的行为都会最大化其收益

  • A self-consistent or stable profile

1-7 Best Response and Nash Equilibrium

Let \(a_{-i} = <a_1, \cdots, a_{i - 1}, a_{i + 1}, \cdots, a_n>\)

then \(a = (a_{-i}, a_i)\)

  • Best Response

    \(a_i^* \in BR(a_{-i}) \iff \forall a_i \in A_i, u_i(a_i^*, a_{-i}) \ge u_i(a_i, a_{-i})\)

  • Nash Equilibrium

    \(a = <a_1, \cdots, a_n>\) is a ("pure strategy") Nash equilibrium iff \(\forall i, a_i \in BR(a_{-i})\)

1-8 Nash Equilibrium of Example Games

1-9 Dominant Strategies

2 strategies \(s_i\), \(s'_i\)

the set of all possible strategy profiles for the other players: \(S_{-i}\)

  • \(s_i\) strictly dominates \(s'_i\)

    if \(\forall s_{-i} \in S_{-i}, u_i(s_i, s_{-i}) \gt u_i(s'_i, s_{-i})\)

  • \(s_i\) very weakly dominates \(s'_i\)

    if \(\forall s_{-i} \in S_{-i}, u_i(s_i, s_{-i}) \ge u_i(s'_i, s_{-i})\)

  • dominant 主导

    If one strategy dominates all others.

  • A strategy profile consisting of dominant strategies for every player must be a Nash equilibrium.

1-10 Pareto Optimality

  • \(o\) Pareto-dominate \(o'\)

    one outcome \(o\) is at least as good for every agent as another outcome \(o'\).

  • \(o^*\) is Pareto Optimality 帕累托最优

    if there is no other outcome that Pareto-dominates it.

Every game has at least 1 Pareto-optimal outcome, and can have more than one Pareto-optimal outcome.