Lecture 51 Introduction to Strategic Interactions

• Strategy = action (or a course) that a person takes when aware about the mutual dependence of results for herself and others.
• Strategic interaction = in which people are aware of ways in which actions affects others
• Depends on one's and others' actions

These models of strategic interaction are called games.

Game Theory

• Each actor knows that their benefits depend on everyone's actions.
• A Game describes
  • players = who interacts with whom
  • feasible strategies = actions available to players
  • information available = what each player knows while making a decision
  • payoffs = outcomes possible for action-combinations (also = Expected utility)
• Understanding interactions based on constraints (to their actions, motives and beliefs about others' actions)
  • self-interest
  • concern for others
  • fairness ke liye preference
• Conflict of interest & Mutual gains
  • Sometimes self-interest can lead to good outcomes for everyone.
    • Limit actions of people for the general good.
  • On the other hand, good for none.

Internalize the effects of our actions and contribute to social goals.

Elements of a Game

• Players
• Strategies
  • Cooperating = taking a risk, opponents may take advantage and reduce our payoff OR in favor of both (Both better off)
  • Defecting = increase own payoff at the expense of the others
• Rule = complete plan of action
  • strategy = specific action or move
• Payoffs
  • Game theory \(\neq\) Decision theory
  • Decision theory \(\to\) Single player's decision matters

Forms of Game

• Normal-form representation
  • move simultaneously
  • clarify strategies, payoffs and players
  • matrix
• Extensive-form
  • don't move simultaneously
  • sequence matters, involves a game tree

Example of an Analytical Game

• "Which tire?" (95 marks)
• The final grade depends on the answer of both the students. Either both get A or both F.
• The two friends are playing a game against each other (though they are not competing against each other)

Nash Equilibrium in Pure Strategies

• Nash Equilibrium = strategy profile \(\implies\) each strategy in the profile is a 'best response' to the other strategies in the profile.
  • Meaning, you can't do any better than this given what other players are up to.
• Nash Equilibria ← each player plays one of the individual strategies available to him or her. (four Nash equilibria, one for each tire)

Pure Coordination Game

• Player's interests are perfectly aligned.
• Let's meet (But don't know which one of the two coffee shops.)
• 0 or 1 utility.

Payoff matrix

$$
\begin{matrix}
& \text{CCD} & \text{Starbucks} \
\text{CCD} & 1,1 & 0,0 \
\text{Starbucks} & 0,0 & 1,1 \
\end{matrix}
$$
There are two NE: \((\text{CCD},\text{CCD})\) and \((\text{Starbucks},\text{Starbucks})\).
- First number is the payoff of player I, strategy is listed in the leftmost column \((\text{first},\text{second})\)

In some coordination games, interests fail to align.

Impure Coordination Game: Battle of Sexes

a.k.a. "Bach or Stravinsky"

• Player I: Woman = concert
  • 2 if both concert,
  • 1 if both theatre
  • 0 if separate places
• Player II: Man = theatre
  • 2 if both theatre,
  • 1 if both concert
  • 0 if separate places

\[ \begin{matrix} & \text{Concert} & \text{Theatre} \\ \text{Concert} & (2,1) & (0,0) \\ \text{Theatre} & (0,0) & (1,2) \end{matrix} \]

Here there are 2 NEs: \((\text{Concert},\text{Concert})\) and \((\text{Theatre},\text{Theatre})\)

But utilities are different (one is more better off than another) \(\implies\) Impure. And one of the player would prefer another, but is worse off if he chooses that.

• No straightforward connection between Nash equilibria and "best" outcome.

Examples of Games & Derivations of NEs

\[ \begin{matrix} & \text{L} & \text{R} \\ \text{U} & (2,2) & (0,0) \\ \text{D} & (0,0) & (1,1) \end{matrix} \]

• \((U,L)\) and \((D,R)\) are 2 NEs
• \((U,L)\) is superior

$$
\begin{matrix}
& \text{L} & \text{R} \
\text{U} & (5,1) & (2,0) \
\text{D} & (5,1) & (1,2)
\end{matrix}
$$
- \((U,L)\) is the only NE
- if player 1 chooses \(U\)
- then player 2 will not choose \(R\) (as it will make him worse off "0")
- \((D,L)\) is not because
- if player 1 chooses \(D\)
- then player 2 will choose \(R\) (he gets 2)

\[ \begin{matrix} & \text{L} & \text{M} & \text{R} \\ \text{U} & (6,2) & (5,1) &\mathbf{(4,3)} \\ \text{M} & (3,6) & (8,4) &(2,1) \\ \text{D} & (2,8) & (9,6) &(3,0) \end{matrix} \]

• (U,R) is the NE though there are better outcomes…
  • (8,4) and (9,6) is better for both
  • BUt If II selects M, I selects D
  • if I selects D, II selects L
  • if II selects L, I selects U
  • if I selects U, II selects R
  • if II selects R, I is better off with U (which he already has selected) so he will stick to U.