The Central Idea
In an ordinary decision problem, the outcome depends on our action and an uncertain state.
In a strategic problem, the outcome depends on our action and the actions of other decision makers.
A strategic decision therefore asks:
What should I do when my best choice depends on what someone else does?
What Is a Game?
In game theory, a game is a mathematical model of interaction among decision makers.
A basic game specifies:
- the players;
- the strategies available to each player;
- the outcomes produced by strategy combinations;
- the payoff or utility each player receives.
Players
The players are the decision makers in the game.
Players may be:
- individual people;
- firms;
- governments;
- teams;
- organizations;
- automated agents.
Strategies
A strategy is a complete plan of action available to a player.
In a one-stage game, a strategy may simply be one available move.
In a multi-stage game, a strategy may specify what the player will do after every relevant future history.
Payoffs
A payoff numerically represents the consequence of a strategy combination for a player.
Higher payoff usually means a more preferred outcome.
Payoffs may represent:
- profit;
- utility;
- market share;
- points;
- cost savings;
- another preference measure.
Payoff Matrices
A two-player game can often be represented by a payoff matrix.
| Player 2: Left | Player 2: Right | |
|---|---|---|
| Player 1: Up | \((3,2)\) | \((1,4)\) |
| Player 1: Down | \((2,1)\) | \((4,3)\) |
In each ordered pair, the first number is Player 1's payoff and the second is Player 2's payoff.
Games Are Not Ordinary Decision Tables
A decision table typically treats columns as states of nature that are not chosen strategically.
In a game, the column player is also choosing.
This means each player must reason about the incentives of the other.
Best Response
A best response is a strategy that gives a player the highest payoff against a specified strategy chosen by the other player.
Best responses are conditional:
If the other player does this, what should I do?
Example: Best Responses
| Player 2: Left | Player 2: Right | |
|---|---|---|
| Player 1: Up | \((4,3)\) | \((1,2)\) |
| Player 1: Down | \((2,1)\) | \((5,4)\) |
If Player 2 chooses Left, Player 1 prefers Up because
\[ 4>2. \]If Player 2 chooses Right, Player 1 prefers Down because
\[ 5>1. \]Dominant Strategy
A dominant strategy gives a player at least as good a payoff as every alternative against every strategy of the other players, and a strictly better payoff in at least one relevant comparison.
If a strategy is strictly better in every comparison, it is a strictly dominant strategy.
Example: Dominant Strategy
| Player 2: Left | Player 2: Right | |
|---|---|---|
| Player 1: Up | \((5,2)\) | \((4,3)\) |
| Player 1: Down | \((2,4)\) | \((1,1)\) |
For Player 1:
\[ 5>2 \]and
\[ 4>1. \]Therefore,
\[ \boxed{\text{Up is strictly dominant for Player 1}}. \]Dominated Strategies
A strategy is dominated if another strategy performs at least as well against every relevant opponent strategy and better against at least one.
Dominated strategies can often be removed from consideration.
Iterated Elimination of Dominated Strategies
In some games, removing one dominated strategy makes another strategy dominated.
Repeating this process is called iterated elimination of dominated strategies.
It can simplify a large strategic problem before equilibrium analysis.
Nash Equilibrium
A Nash equilibrium is a strategy combination in which each player's strategy is a best response to the strategies chosen by the other players.
At a Nash equilibrium, no player can improve their own payoff by changing strategy alone.
Example: Find a Nash Equilibrium
| Player 2: Left | Player 2: Right | |
|---|---|---|
| Player 1: Up | \((3,3)\) | \((0,4)\) |
| Player 1: Down | \((4,0)\) | \((1,1)\) |
For Player 1, Down is better whether Player 2 chooses Left or Right.
For Player 2, Right is better whether Player 1 chooses Up or Down.
Therefore,
\[ \boxed{(\text{Down},\text{Right})} \]is a Nash equilibrium.
Equilibrium Does Not Mean Best for Everyone
A Nash equilibrium is stable against unilateral deviation.
It does not necessarily maximize:
- total payoff;
- fairness;
- social welfare;
- the payoff of every player.
The Prisoner's Dilemma Pattern
A famous strategic structure occurs when each player has an individually attractive action that leads to a worse joint result.
| Player 2: Cooperate | Player 2: Defect | |
|---|---|---|
| Player 1: Cooperate | \((3,3)\) | \((0,5)\) |
| Player 1: Defect | \((5,0)\) | \((1,1)\) |
Why Defection Is Dominant
If Player 2 cooperates, Player 1 prefers Defect because
\[ 5>3. \]If Player 2 defects, Player 1 still prefers Defect because
\[ 1>0. \]The same reasoning applies symmetrically to Player 2.
Prisoner's Dilemma Equilibrium
Both players choose Defect, producing
\[ (1,1). \]Yet mutual cooperation would produce
\[ (3,3). \]The equilibrium is individually stable but jointly inferior.
Coordination Games
A coordination game rewards players for choosing compatible actions.
| Player 2: A | Player 2: B | |
|---|---|---|
| Player 1: A | \((4,4)\) | \((0,0)\) |
| Player 1: B | \((0,0)\) | \((3,3)\) |
Both
\[ (A,A) \]and
\[ (B,B) \]are Nash equilibria.
Multiple Equilibria
Some games have more than one Nash equilibrium.
When this occurs, equilibrium analysis alone may not predict which outcome will be selected.
History, communication, conventions, expectations, and institutions may influence equilibrium selection.
Anti-Coordination Games
In an anti-coordination game, players prefer to choose different actions.
Such structures arise when crowding, congestion, specialization, or role differentiation matters.
Zero-Sum Games
A zero-sum game is one in which one player's gain is exactly the other player's loss.
If Player 1's payoff is \(u\), Player 2's payoff is
\[ -u. \]The payoffs sum to
\[ \boxed{0}. \]Payoff Matrix for a Zero-Sum Game
Since one player's payoff determines the other's, a zero-sum matrix can be written using only Player 1's payoffs:
| Player 2: Left | Player 2: Right | |
|---|---|---|
| Player 1: Up | 3 | -1 |
| Player 1: Down | 0 | 2 |
Maximin in Zero-Sum Games
Player 1 can protect against the worst response by choosing the strategy with the largest minimum payoff.
This is the maximin criterion:
\[ \boxed{ \max_i\min_j a_{ij} }. \]Minimax in Zero-Sum Games
Player 2 can minimize Player 1's best possible payoff.
\[ \boxed{ \min_j\max_i a_{ij} }. \]When maximin equals minimax in pure strategies, the game has a saddle point.
Saddle Point
A saddle point is an entry that is simultaneously:
- the minimum in its row for the maximizing player;
- the maximum in its column for the minimizing player.
At such a point,
\[ \max\min=\min\max. \]Mixed Strategies
A mixed strategy randomizes among pure strategies.
If Player 1 chooses Up with probability \(p\) and Down with probability \(1-p\), then
\[ \boxed{ P(\text{Up})=p,\qquad P(\text{Down})=1-p }. \]Why Randomization Can Be Strategic
If an opponent can exploit a predictable pure strategy, randomization may prevent that exploitation.
Mixed strategies are especially important in competitive settings where unpredictability itself has value.
Expected Payoff under a Mixed Strategy
Suppose Player 1 chooses Up with probability \(p\) and Down with probability \(1-p\).
Against a fixed opponent strategy, expected payoff is the probability-weighted average of the pure-strategy payoffs.
Example: Expected Payoff from Mixing
Suppose Up pays \(6\) against Left and Down pays \(2\) against Left.
If Player 1 uses Up with probability \(p\), expected payoff against Left is
\[ 6p+2(1-p). \]Simplify:
\[ \boxed{2+4p}. \]Indifference Method for Mixed Strategies
In many \(2\times2\) games, an equilibrium mixing probability is found by making the opponent indifferent between their pure strategies.
If one pure strategy were strictly better, the opponent would not mix.
Example: Making the Opponent Indifferent
Suppose Player 1's mixing probability \(p\) causes Player 2's expected payoff from Left to be
\[ 1+3p \]and from Right to be
\[ 4-2p. \]Set them equal:
\[ 1+3p=4-2p. \] \[ 5p=3. \] \[ \boxed{p=\frac35}. \]Mixed-Strategy Equilibrium
In a mixed-strategy Nash equilibrium, each player's randomization makes the other player willing to randomize among the strategies used with positive probability.
Expected payoffs, rather than deterministic outcomes, determine the equilibrium conditions.
Simultaneous Games
In a simultaneous game, players choose without observing the other players' current choices.
Payoff matrices are especially natural for these games.
Sequential Games
In a sequential game, some players act after observing earlier moves.
Game trees are often more useful than matrices because the order of moves matters.
Backward Induction
Sequential games are often solved using backward induction.
- Begin at the final decision nodes.
- Determine the best action there.
- Replace those future branches with their resulting values.
- Continue moving backward toward the start.
Credible Strategies
In sequential games, a threatened future action matters only if carrying it out would actually be optimal when that future decision point is reached.
This leads to the idea of credibility.
A threat that would never rationally be carried out may fail to influence earlier behavior.
Repeated Games
A repeated game occurs when the same strategic interaction takes place more than once.
Future consequences can change incentives today.
Cooperation may become sustainable when current actions affect future treatment.
Discounting Future Payoffs
If future payoffs are discounted by factor
\[ 0<\delta<1, \]a payoff \(x\) received one period later may be valued as
\[ \delta x. \]Larger \(\delta\) means future consequences matter more strongly.
Cooperation in Repeated Interaction
A player may sacrifice a short-run gain to preserve larger future benefits.
This means repeated interaction can produce strategic behavior that would not arise in a one-shot game.
Incomplete Information
In some games, players do not know everything about one another.
They may be uncertain about:
- costs;
- preferences;
- resources;
- available strategies;
- private information.
Probability models can be used to represent these beliefs.
Strategic Information
Information in a game can affect behavior, not merely prediction.
Revealing information may cause another player to change strategy.
This makes the value of information in strategic settings more subtle than in ordinary one-person decision problems.
Commitment
A commitment changes future options in a way that can alter the incentives of other players.
Giving up flexibility can sometimes improve a strategic position if the commitment is observable and credible.
First-Mover and Second-Mover Effects
Acting first can be advantageous when commitment matters.
Acting second can be advantageous when observing the first move provides useful information.
There is no universal rule that moving first or second is always better.
Pareto Improvement
An outcome is a Pareto improvement over another outcome if at least one player is better off and no player is worse off.
This provides a way to compare joint outcomes without adding the players' utilities together.
Pareto Efficiency
An outcome is Pareto efficient if no feasible alternative makes at least one player better off without making another player worse off.
A Nash equilibrium need not be Pareto efficient.
Game Theory and Incentive Design
Strategic analysis is not only about predicting choices.
It can also be used to design:
- contracts;
- rules;
- auctions;
- rewards;
- penalties;
- institutions.
Changing the payoff structure can change strategic behavior.
Common Error: Choosing the Largest Number in the Matrix
A player cannot simply choose the globally largest payoff entry.
The realized cell depends on what the other player chooses.
Strategic reasoning requires comparing payoffs conditional on opponent actions.
Common Error: Confusing Dominant Strategy with Best Response
A best response is optimal against one specified opponent strategy.
A dominant strategy is optimal across all relevant opponent strategies.
Common Error: Assuming Every Game Has a Pure-Strategy Equilibrium
Some finite games have no Nash equilibrium in pure strategies.
Mixed strategies may be required.
Common Error: Assuming Equilibrium Means Cooperation
Equilibrium means no profitable unilateral deviation.
It does not mean the players are cooperating or achieving the best joint outcome.
Common Error: Ignoring Sequence
A matrix can hide important timing information in a sequential game.
When players observe earlier moves, game-tree structure and backward induction may be necessary.
Common Error: Treating Randomization as Indecision
In game theory, a mixed strategy can be intentionally optimal.
Randomization can prevent an opponent from exploiting predictable behavior.
A Reliable Procedure for Two-Player Payoff Matrices
- Identify the players and their strategies.
- Read the payoff order correctly.
- Find each player's best response to every opponent strategy.
- Check for dominant and dominated strategies.
- Look for cells where both players are playing best responses.
- Identify pure-strategy Nash equilibria.
- If none exist, consider mixed strategies.
- Interpret the equilibrium relative to the broader objectives of the game.
A Reliable Procedure for Sequential Games
- Draw the order of moves explicitly.
- Identify which player moves at each decision node.
- Record terminal payoffs.
- Start at the final decision nodes.
- Choose the optimal continuation there.
- Work backward through earlier decisions.
- Check whether threats or promises are credible.
- Interpret how timing changes strategic incentives.
Practice 1: Best Response
| Player 2: Left | Player 2: Right | |
|---|---|---|
| Player 1: Up | \((5,2)\) | \((1,4)\) |
| Player 1: Down | \((3,1)\) | \((6,3)\) |
What is Player 1's best response to Left? What is Player 1's best response to Right?
Practice 2: Dominant Strategy
| Player 2: L | Player 2: R | |
|---|---|---|
| Player 1: U | \((4,2)\) | \((5,1)\) |
| Player 1: D | \((2,3)\) | \((1,4)\) |
Does Player 1 have a dominant strategy?
Practice 3: Nash Equilibrium
| Player 2: L | Player 2: R | |
|---|---|---|
| Player 1: U | \((3,3)\) | \((0,2)\) |
| Player 1: D | \((2,0)\) | \((1,1)\) |
Identify the pure-strategy Nash equilibrium.
Practice 4: Coordination
| Player 2: A | Player 2: B | |
|---|---|---|
| Player 1: A | \((5,5)\) | \((0,0)\) |
| Player 1: B | \((0,0)\) | \((4,4)\) |
How many pure-strategy Nash equilibria are there?
Practice 5: Zero-Sum Maximin
| L | R | |
|---|---|---|
| U | 4 | 1 |
| D | 2 | 3 |
Find Player 1's maximin value.
Practice 6: Zero-Sum Minimax
Using the same matrix, find Player 2's minimax value for Player 1's payoff.
Practice 7: Mixed Expected Payoff
Player 1 chooses Up with probability \(0.7\) and Down with probability \(0.3\).
Against a fixed opponent move, Up pays \(8\) and Down pays \(2\).
Find Player 1's expected payoff.
Practice 8: Indifference Probability
An opponent's expected payoff from Left is
\[ 2+4p \]and from Right is
\[ 6-2p. \]Find the value of \(p\) that makes the opponent indifferent.
Practice 9: Repeated Game Discounting
A payoff of \(100\) is received one period later and the discount factor is
\[ \delta=0.8. \]What is its discounted value?
Practice 10: Equilibrium versus Efficiency
A Nash equilibrium gives payoffs \((2,2)\), while another feasible outcome gives \((4,4)\).
Is the equilibrium Pareto efficient?
Practice Answers
Practice 1
Against Left, Player 1 compares \(5\) with \(3\), so the best response is Up.
Against Right, Player 1 compares \(1\) with \(6\), so the best response is Down.
Practice 2
Player 1 compares:
\[ 4>2 \]against L and
\[ 5>1 \]against R.
Therefore,
\[ \boxed{U\text{ is strictly dominant}}. \]Practice 3
At \((U,L)\), Player 1 does not want to switch to D, and Player 2 does not want to switch to R.
\[ \boxed{(U,L)} \]is the pure-strategy Nash equilibrium.
Practice 4
Both coordinated outcomes are equilibria:
\[ \boxed{(A,A)\text{ and }(B,B)}. \]Therefore there are
\[ \boxed{2} \]pure-strategy Nash equilibria.
Practice 5
Row minima are
\[ \min(4,1)=1 \]and
\[ \min(2,3)=2. \]Therefore,
\[ \boxed{\max\{1,2\}=2}. \]Practice 6
Column maxima are
\[ \max(4,2)=4 \]and
\[ \max(1,3)=3. \]Therefore,
\[ \boxed{\min\{4,3\}=3}. \]Since \(2\ne3\), there is no pure-strategy saddle point.
Practice 7
\[ E[X] = 0.7(8)+0.3(2) \] \[ =5.6+0.6 \] \[ \boxed{6.2}. \]Practice 8
Set the expected payoffs equal:
\[ 2+4p=6-2p. \] \[ 6p=4. \] \[ \boxed{p=\frac23}. \]Practice 9
\[ 0.8(100)=\boxed{80}. \]Practice 10
No. The outcome \((4,4)\) makes both players better off than \((2,2)\).
Therefore the equilibrium is
\[ \boxed{\text{not Pareto efficient}}. \]What to Remember
Strategic decision problems differ from ordinary uncertainty because other decision makers actively choose actions that affect our outcome.
- Games specify players, strategies, and payoffs.
- Best responses are conditional on what others do.
- Dominant strategies remain best across opponent choices.
- Nash equilibrium is mutual best response.
- An equilibrium need not maximize joint welfare.
- Coordination games can have multiple equilibria.
- Zero-sum games emphasize maximin and minimax reasoning.
- Mixed strategies deliberately randomize among pure strategies.
- Sequential games are often analyzed by backward induction.
- Repeated interaction can change incentives through future consequences.
The core strategic principle is: the best action depends not only on what can happen, but also on what other decision makers have incentives to do.
Where We Go Next
Game theory studies decisions whose outcomes depend on other decision makers. The next lesson turns to a different multi-choice problem: allocating limited resources across several opportunities under uncertainty and competing objectives.
The next lesson, Portfolio and Allocation, develops diversification, weighted returns, constraints, tradeoffs, and allocation decisions across multiple assets, projects, or uses of resources.