Unit 8: Game Theory - Subjective Questions

Game theory is the study of strategic decision-making among rational, interdependent players whose outcomes depend not only on their own choices but also on the choices of others.

Key elements of a game:

• Players: The decision-makers (firms, individuals, governments).
• Strategies: The complete plan of actions available to each player.
• Payoffs: The outcomes (profit, utility) resulting from a combination of strategies.
• Information: What each player knows when making decisions.

Significance in managerial economics:

• Helps analyze oligopoly behavior where a few firms are mutually interdependent.
• Aids in pricing, advertising, and output decisions under competition.
• Supports strategic planning such as entry/exit and capacity choices.
• Explains phenomena like cartels, price wars, and cooperation.

Game theory equips managers to anticipate rivals' reactions and choose strategies that maximize their own payoffs.

Games are classified along several dimensions:

1. Cooperative vs. Non-cooperative games

• Cooperative: Players can form binding agreements (e.g., cartels like OPEC).
• Non-cooperative: No enforceable agreements; each player acts independently (e.g., price competition).

2. Zero-sum vs. Non-zero-sum games

• Zero-sum: One player's gain equals another's loss; total payoff is constant (e.g., poker).
• Non-zero-sum: Total payoffs can vary; mutual gain or loss is possible (e.g., prisoner's dilemma).

3. Simultaneous vs. Sequential games

• Simultaneous: Players move at the same time without knowing others' choices (e.g., sealed bids).
• Sequential: Players move in turn, observing prior moves (e.g., chess, entry deterrence).

4. Symmetric vs. Asymmetric games

• Symmetric: Payoffs depend only on strategies, not on who plays them.
• Asymmetric: Players have different strategy sets or payoffs.

5. One-shot vs. Repeated games

• One-shot: Played only once.
• Repeated: Played multiple times, allowing reputation and cooperation to develop.

A dominant strategy is one that yields a player the best payoff regardless of what the other players choose. A rational player will always play a dominant strategy if one exists.

Strictly dominant strategy:

• Gives a strictly higher payoff than any other strategy against all opponent choices.
• If strategy strictly dominates , then payoff payoff for every combination of others' actions.

Weakly dominant strategy:

• Gives a payoff at least as high as any other strategy, and strictly higher for at least one opponent choice.
• If weakly dominates : payoff payoff always, with strict inequality in at least one case.

Example: In the prisoner's dilemma, "Confess" is a strictly dominant strategy for both players because confessing always yields a lower sentence regardless of the other player's action.

Dominant strategy equilibrium occurs when every player has a dominant strategy and plays it.

Nash equilibrium (NE) is a set of strategies, one for each player, such that no player can improve their payoff by unilaterally changing their own strategy while the others keep theirs unchanged.

Formally, a strategy profile is a Nash equilibrium if for every player :

where denotes the equilibrium strategies of all other players.

Difference from dominant strategy equilibrium:

• A dominant strategy equilibrium requires each player's chosen strategy to be best against all possible opponent strategies.
• A Nash equilibrium requires each player's strategy to be best only against the specific equilibrium strategies of the others.

Key points:

• Every dominant strategy equilibrium is a Nash equilibrium, but not vice versa.
• A game may have no, one, or multiple Nash equilibria.
• NE captures mutual best responses even when no dominant strategy exists.

The Prisoner's Dilemma illustrates why two rational individuals may fail to cooperate even when cooperation yields a better joint outcome.

Scenario: Two suspects are interrogated separately. Each can Confess or Stay Silent. Payoffs (years in prison, lower is better):

(Values represent utility; higher is better, so fewer years = higher payoff.)

Analysis:

• For each player, Confess is a strictly dominant strategy:
  • If the other stays silent, confessing gives vs. .
  • If the other confesses, confessing gives vs. .
• Both confess, reaching the Nash equilibrium (Confess, Confess) with payoff .

Why it is inefficient:

• The cooperative outcome (Silent, Silent) = is better for both.
• Individual rationality leads to a collectively worse outcome.
• This explains cartel breakdowns, price wars, and free-riding in economics.

Pure strategy: A player chooses one specific action with certainty (probability = 1).

Mixed strategy: A player randomizes over two or more pure strategies according to a probability distribution. For example, playing strategy with probability and with probability .

Differences:

Why mixed strategies matter:

• Some games (like Matching Pennies) have no pure-strategy Nash equilibrium.
• Nash's theorem guarantees that every finite game has at least one equilibrium in mixed strategies.
• Randomization prevents opponents from exploiting predictable behavior (e.g., a penalty taker mixing left/right shots).

Mixed-strategy Nash equilibrium occurs where each player's mix makes the opponent indifferent among their own strategies.

Cooperative games:

• Players can form binding and enforceable agreements.
• Focus is on how coalitions form and how joint payoffs are shared.
• Example: A cartel like OPEC where members agree to restrict output to raise prices; a joint venture between firms.

Non-cooperative games:

• No binding agreements are possible; each player acts in self-interest.
• Focus is on individual strategies and equilibrium (e.g., Nash equilibrium).
• Example: Two firms independently setting prices in a competitive market; bidding in an auction.

Comparison table:

Most oligopoly analysis uses non-cooperative game theory because binding agreements (collusion) are often illegal or hard to enforce.

Iterated elimination of dominated strategies (IEDS) is a solution technique that simplifies a game by repeatedly removing strategies that are dominated (never a best choice), narrowing down to a likely outcome.

Procedure:

1. Identify a strictly dominated strategy for any player and delete it.
2. In the reduced game, look for new dominated strategies.
3. Repeat until no more strategies can be eliminated.

Example: Consider payoffs for Player A (rows) and B (columns):

• For Player A, strategy M is dominated by D (2 > 1 and 2 > 1). Eliminate M.
• Comparing U and D for A: against L, U=3>D=2; against R, D=2>U=0. Neither dominates, so check B.
• For Player B, L dominates R (1≥0 and 2≥1). Eliminate R.
• With only L left, Player A picks U (3 > 2).

Result: Predicted outcome is (U, L) = .

Note: Eliminating strictly dominated strategies preserves all Nash equilibria; eliminating weakly dominated ones may remove some.

In an oligopoly, a few firms are interdependent, so each firm's optimal price depends on rivals' pricing. Game theory models this strategic interaction.

Pricing dilemma (a prisoner's dilemma in disguise): Two firms choose between High Price (collude) and Low Price (compete). Payoffs = profits (in millions):

Analysis:

• For each firm, Low Price is a dominant strategy:
  • If rival prices High, undercutting gives 70 > 50.
  • If rival prices Low, matching gives 30 > 20.
• Nash equilibrium is (Low, Low) = .

Insight:

• Joint profit is maximized at (High, High) = , but the temptation to undercut breaks cooperation.
• This explains why cartels are unstable and price wars erupt.
• In repeated games, firms may sustain collusion through trigger strategies and threats of retaliation.

Simultaneous games:

• Players move at the same time or without observing the other's move.
• Represented using the normal (strategic) form — a payoff matrix.
• Solution: Nash equilibrium.
• Example: Two firms simultaneously choosing output (Cournot competition).

Sequential games:

• Players move in a fixed order; later movers observe earlier moves.
• Timing and information matter; a first-mover advantage may exist.
• Represented using the extensive form — a game tree.
• Solution: subgame perfect Nash equilibrium found by backward induction.
• Example: A firm deciding whether to enter a market, then the incumbent deciding to fight or accommodate.

Game tree components:

• Nodes: Decision points for players.
• Branches: Available actions.
• Terminal nodes: Final outcomes with payoffs.

Backward induction: Solve from the last decisions backward, ensuring each choice is optimal at every stage, eliminating non-credible threats.

To find pure-strategy Nash equilibria, identify each player's best responses.

Player A's best responses (compare rows for each column):

• If B plays Left: Up gives 4, Down gives 2 → best is Up.
• If B plays Right: Up gives 1, Down gives 3 → best is Down.

Player B's best responses (compare columns for each row):

• If A plays Up: Left gives 3, Right gives 2 → best is Left.
• If A plays Down: Left gives 1, Right gives 4 → best is Right.

Check each cell for mutual best response:

• (Up, Left) = (4, 3): A best-responds with Up ✓, B best-responds with Left ✓ → Nash equilibrium.
• (Down, Right) = (3, 4): A best-responds with Down ✓, B best-responds with Right ✓ → Nash equilibrium.

Result: The game has two pure-strategy Nash equilibria: and . This resembles a coordination game where both players prefer to align their choices.

Matching Pennies is a zero-sum game with no pure-strategy Nash equilibrium, so we solve for the mixed-strategy equilibrium.

Setup: Let Player A play Heads with probability and Tails with . Let Player B play Heads with probability and Tails with .

Make B indifferent (to find ): B's expected payoffs must be equal.

• B's payoff from Heads:
• B's payoff from Tails:

Set equal:

Make A indifferent (to find ): A's expected payoffs must be equal.

• A's payoff from Heads:
• A's payoff from Tails:

Set equal:

Mixed-strategy Nash equilibrium: Each player plays Heads and Tails with probability . The expected payoff to each player is .

A zero-sum game is one in which the total payoff to all players is constant (zero) for every outcome. One player's gain is exactly balanced by another's loss.

Mathematically, for players A and B: .

Comparison:

• Zero-sum game: Sum of payoffs = 0. Pure conflict; no room for cooperation.
  • Example: Poker, dividing a fixed prize, Matching Pennies.
• Positive-sum game: Sum of payoffs > 0 in some outcomes. Cooperation can make all players better off.
  • Example: Trade between countries, joint ventures.
• Negative-sum game: Sum of payoffs < 0. All players may end up worse off.
  • Example: Costly price wars, arms races, prolonged litigation.

Managerial relevance: Recognizing whether a situation is zero-sum or positive-sum helps managers decide between competition and cooperation. Treating a positive-sum situation as zero-sum can destroy mutual value.

Backward induction is a method for solving sequential games by reasoning from the end of the game to the beginning. Players anticipate future optimal responses and choose accordingly, yielding the subgame perfect Nash equilibrium.

Market entry example:

• Stage 1: A potential Entrant decides to Enter or Stay Out.
• Stage 2: If entry occurs, the Incumbent decides to Fight (price war) or Accommodate.

Payoffs (Entrant, Incumbent):

• Stay Out:
• Enter, Incumbent Fights:
• Enter, Incumbent Accommodates:

Backward induction:

1. Analyze Stage 2: If entry occurs, Incumbent compares Fight () vs. Accommodate (). It chooses Accommodate (5 > 2).
2. Analyze Stage 1: Knowing the Incumbent will accommodate, the Entrant compares Enter () vs. Stay Out (). It chooses Enter.

Equilibrium outcome: Enter, Accommodate with payoff .

Key insight: A threat to "Fight" is a non-credible threat because the Incumbent would not actually carry it out. Backward induction eliminates such empty threats.

The Battle of the Sexes is a coordination game where two players prefer to be together but disagree on the preferred activity.

Scenario: A couple decides between attending a Football match or an Opera. Both prefer being together over being apart, but each has a favorite event.

Payoff matrix (Man, Woman):

Finding pure-strategy Nash equilibria:

• (Football, Football) = (2, 1): Neither gains by deviating (deviating gives 0) → NE.
• (Opera, Opera) = (1, 2): Neither gains by deviating → NE.

Mixed-strategy equilibrium also exists: Each player randomizes; the man plays Football with probability and the woman plays Opera with probability (derived from indifference conditions).

Insight: The game highlights the coordination problem — multiple equilibria exist, and players need communication or conventions to select one. It applies to standard-setting and technology-adoption decisions in business.

In a one-shot game like the prisoner's dilemma, rational players defect. However, when the game is repeated, cooperation can be sustained because players value future payoffs and fear retaliation.

Why cooperation emerges in repeated games:

• Players interact multiple times, so today's actions affect future outcomes.
• The threat of future punishment deters cheating.
• This applies to firms competing repeatedly on price or output.

Trigger strategies:

A trigger strategy prescribes cooperation as long as everyone cooperates, but switches to punishment if anyone defects.

• Grim trigger: Cooperate until a rival defects, then defect forever.
• Tit-for-tat: Start by cooperating, then copy the rival's previous move.

Condition for cooperation: Cooperation is sustainable if the present value of cooperating exceeds the one-time gain from cheating plus the punishment losses. This depends on the discount factor : cooperation holds when players value the future highly (high ).

Managerial application: Explains how oligopolists sustain tacit collusion and stable prices without explicit agreements. In a finitely repeated game with a known end, backward induction may unravel cooperation.

Both are types of Nash equilibrium, but they differ in how players choose their actions.

Pure-strategy Nash equilibrium (PSNE):

• Each player selects a single, definite action with certainty.
• The chosen actions are mutual best responses.
• May not exist in some games (e.g., Matching Pennies).
• Example: (Confess, Confess) in the prisoner's dilemma.

Mixed-strategy Nash equilibrium (MSNE):

• Each player randomizes over strategies using a probability distribution.
• Each player's probabilities make the opponent indifferent among their options.
• Always exists in finite games (Nash's existence theorem).
• Example: Playing Heads and Tails each with probability in Matching Pennies.

Comparison table:

A game may have only pure equilibria, only mixed, or both.

First-mover advantage arises in sequential games when the player who moves first can influence the outcome favorably by committing to an action before rivals respond.

How it works:

• By moving first, a firm can credibly commit to a strategy (e.g., large capacity or aggressive output).
• This constrains the rival's best response, often to the first mover's benefit.
• Example (Stackelberg model): The leader firm sets output first, capturing a larger market share and higher profit than the follower.

Illustration: In a quantity game, the leader produces a large quantity, forcing the follower to produce less. Commitment must be credible (hard to reverse) to work.

Second-mover advantage exists when moving later is better:

• Learning from the first mover's mistakes or market response.
• Free-riding on the first mover's R&D, advertising, or market development.
• Avoiding the cost of establishing a new product or market.
• Example: A firm that waits and copies a proven product design, avoiding pioneering costs.

Conclusion: Whether first or second moving is advantageous depends on commitment value versus information and cost-saving benefits.

We analyze each firm's best strategy. Payoffs represent profits.

Player A's dominant strategy:

• If B Advertises: A gets 30 (Advertise) vs. 20 (Don't) → prefer Advertise.
• If B Doesn't: A gets 60 (Advertise) vs. 50 (Don't) → prefer Advertise.
• Advertise strictly dominates Don't for A.

Player B's dominant strategy (symmetric):

• If A Advertises: B gets 30 vs. 20 → prefer Advertise.
• If A Doesn't: B gets 60 vs. 50 → prefer Advertise.
• Advertise strictly dominates Don't for B.

Nash equilibrium: Both play their dominant strategy → (Advertise, Advertise) = .

Efficiency comment:

• The equilibrium payoff is worse than the mutual outcome (Don't, Don't) = .
• This is a prisoner's dilemma structure: individually rational advertising leads to a collectively inferior result because advertising costs cancel out competitively.
• Both firms would be better off with a mutual agreement not to advertise, but each has an incentive to deviate. This explains "advertising wars" in industries like soft drinks and telecom.

While game theory is a powerful analytical tool, it has several limitations when applied to real business decisions:

1. Assumption of rationality:

• Assumes all players are perfectly rational and payoff-maximizing.
• Real managers exhibit bounded rationality, biases, and emotions.

2. Complete information assumption:

• Many models assume players know all strategies and payoffs.
• In reality, information is incomplete and asymmetric.

3. Difficulty measuring payoffs:

• Assigning precise numerical payoffs to real-world outcomes is often impractical.

4. Multiple equilibria problem:

• Some games have several Nash equilibria with no clear rule for which will occur, limiting predictive power.

5. Static nature:

• Simple models ignore the dynamic, evolving nature of markets and changing strategies.

6. Ignores communication and trust:

• Non-cooperative models understate the role of negotiation, reputation, and relationships.

7. Complexity:

• Real markets involve many players and variables, making models hard to solve or apply.

Conclusion: Despite these limits, game theory provides valuable qualitative insights into strategic interdependence, helping managers structure their thinking even when exact predictions are impossible.