Games and Strategic Behavior | Malone Math

The Central Idea

In an ordinary optimization problem, one decision maker chooses an action and evaluates the resulting outcome.

In a strategic problem, the outcome also depends on choices made by others.

The central idea is:

\[ \boxed{\text{my best choice may depend on your choice}} \]

Game theory studies these interdependent decisions.

The Basic Ingredients of a Game

A mathematical game usually specifies:

players — the decision makers;
strategies — the actions available to each player;
payoffs — the value each player receives from each combination of strategies;
information — what each player knows when choosing;
rules — how and when decisions are made.

Players

A player is any decision-making entity whose choices affect the outcome.

Players may be:

individual people;
firms;
governments;
teams;
software agents;
organizations.

Strategies

A strategy is a rule or action available to a player.

In a simple one-stage game, strategies may be ordinary actions such as:

cooperate or defect;
enter or stay out;
set a high price or low price;
advertise or do not advertise.

In multi-stage games, a strategy may specify what a player will do after every relevant history or information state.

Payoffs

A payoff assigns a numerical value to an outcome from a player's perspective.

Payoffs may represent:

profit;
cost;
utility;
market share;
points;
probability of success;
another ordered measure of preference.

Larger numbers usually represent more preferred outcomes unless the model states otherwise.

Payoff Matrices

A two-player game with a small number of strategies can be represented by a payoff matrix.

Suppose Player 1 chooses rows and Player 2 chooses columns.

	Player 2: Left	Player 2: Right
Player 1: Up	\((3,2)\)	\((1,4)\)
Player 1: Down	\((5,1)\)	\((2,3)\)

In each ordered pair,

\[ \boxed{(\text{Player 1 payoff},\text{Player 2 payoff})}. \]

Reading a Payoff Matrix

If Player 1 chooses Up and Player 2 chooses Left, the outcome is

\[ (3,2). \]

Player 1 receives payoff \(3\).

Player 2 receives payoff \(2\).

If Player 1 chooses Down and Player 2 chooses Right, the outcome is

\[ (2,3). \]

Best Responses

A best response is a strategy that gives a player the highest payoff given the other player's strategy.

Consider Player 1 in the previous matrix.

If Player 2 chooses Left:

\[ \text{Up gives }3,\qquad \text{Down gives }5. \]

Therefore, Down is Player 1's best response to Left.

If Player 2 chooses Right:

\[ \text{Up gives }1,\qquad \text{Down gives }2. \]

Again, Down is Player 1's best response.

Dominant Strategies

A dominant strategy gives a player at least as good a payoff as every alternative strategy for every possible strategy choice by the other player, and strictly better somewhere under the standard strict dominance comparison.

In the example, Down gives Player 1:

\[ 5>3 \]

when Player 2 chooses Left, and

\[ 2>1 \]

when Player 2 chooses Right.

Therefore, Down strictly dominates Up.

Dominated Strategies

A dominated strategy is always worse than another strategy under the relevant comparison.

If a strategy is strictly dominated, a rational payoff-maximizing player has no reason to choose it in the one-shot game under the model's assumptions.

Removing dominated strategies can simplify a game.

Iterated Elimination

Sometimes removing one dominated strategy creates another dominance relationship.

The process of repeatedly removing dominated strategies is called iterated elimination of dominated strategies.

The exact conclusions can depend on whether weak or strict dominance is used.

Nash Equilibrium

A Nash equilibrium is a strategy profile in which no player can improve their payoff by changing only their own strategy while the other players keep theirs fixed.

Symbolically, a profile \(s^\ast\) is a Nash equilibrium when each player's strategy is a best response to the strategies of the others.

Mutual Best Responses

In a two-player payoff matrix, a pure-strategy Nash equilibrium occurs at a cell where:

the row player's strategy is a best response to the chosen column;
the column player's strategy is a best response to the chosen row.

Therefore, Nash equilibria can often be found by marking best responses.

A Nash-Equilibrium Example

	Left	Right
Up	\((4,3)\)	\((1,2)\)
Down	\((2,1)\)	\((3,4)\)

Against Left, Player 1 prefers Up because

\[ 4>2. \]

Against Up, Player 2 prefers Left because

\[ 3>2. \]

Therefore,

\[ \boxed{(\text{Up},\text{Left})} \]

is a Nash equilibrium.

Similarly,

\[ \boxed{(\text{Down},\text{Right})} \]

is also a Nash equilibrium.

Multiple Equilibria

A game can have more than one Nash equilibrium.

When several equilibria exist, the model may need additional reasoning about:

expectations;
communication;
conventions;
risk;
history;
equilibrium selection.

No Pure-Strategy Equilibrium

Some games have no Nash equilibrium in pure strategies.

A classic example has players trying to mismatch or match actions in a way that causes each player to want to change after seeing the other player's fixed choice.

Such games motivate mixed strategies.

Mixed Strategies

A mixed strategy assigns probabilities to pure strategies.

If Player 1 chooses Up with probability \(p\) and Down with probability \(1-p\), then

\[ \boxed{0\le p\le1}. \]

The player is not choosing both actions simultaneously. The probabilities describe a randomized decision rule.

Expected Payoffs Under Mixing

Suppose an action yields payoff \(a\) with probability \(p\) and payoff \(b\) with probability \(1-p\).

The expected payoff is

\[ \boxed{pa+(1-p)b}. \]

Mixed-strategy analysis uses expected payoffs to identify probabilities that make opponents indifferent among strategies they use with positive probability.

Indifference Principle

In a fully mixed equilibrium, each player chooses probabilities that make the other player indifferent between the pure strategies used in the mix.

If one option gave the opponent a strictly larger expected payoff, the opponent would not randomize between them.

A Simple Mixed-Strategy Calculation

Suppose Player 2 chooses Left with probability \(q\) and Right with probability \(1-q\).

Player 1's payoff from Up is

\[ U_{\text{Up}}=4q+0(1-q)=4q. \]

Player 1's payoff from Down is

\[ U_{\text{Down}}=1q+3(1-q). \]

Simplify:

\[ U_{\text{Down}}=3-2q. \]

To make Player 1 indifferent,

\[ 4q=3-2q. \]

Therefore,

\[ 6q=3 \]

and

\[ \boxed{q=\frac12}. \]

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 receives payoff \(x\), Player 2 receives

\[ -x. \]

Therefore, the payoffs sum to

\[ x+(-x)=\boxed{0}. \]

One-Payoff Representation

Because Player 2's payoff is determined by Player 1's in a zero-sum game, the game can be represented by a single payoff matrix for one player.

For example,

\[ \begin{bmatrix} 2 & -1\\ -3 & 4 \end{bmatrix} \]

may represent Player 1's payoffs.

Player 2 receives the negatives of these values.

Maximin Reasoning

In a zero-sum game, Player 1 may examine the minimum payoff guaranteed by each row.

For

\[ \begin{bmatrix} 2 & -1\\ -3 & 4 \end{bmatrix}, \]

the row minima are

\[ -1,\qquad -3. \]

Player 1's maximin value is

\[ \boxed{\max(-1,-3)=-1}. \]

Minimax Reasoning

Player 2 tries to minimize Player 1's payoff.

For the same matrix, the column maxima are

\[ 2,\qquad4. \]

Player 2's minimax value is

\[ \boxed{\min(2,4)=2}. \]

Saddle Points

A saddle point exists in a zero-sum matrix game when the row player's maximin value equals the column player's minimax value.

In that case, a pure-strategy equilibrium exists.

\[ \text{maximin}<\text{minimax}, \]

mixed strategies may be required.

The Prisoner's Dilemma Structure

One famous strategic pattern occurs when each player has an individual incentive to defect even though mutual cooperation would make both better off than mutual defection.

	Cooperate	Defect
Cooperate	\((3,3)\)	\((0,5)\)
Defect	\((5,0)\)	\((1,1)\)

Defect strictly dominates Cooperate for each player.

Therefore,

\[ \boxed{(\text{Defect},\text{Defect})} \]

is the one-shot Nash equilibrium even though

\[ (3,3) \]

is better for both than

\[ (1,1). \]

Individual and Collective Incentives

The Prisoner's Dilemma illustrates an important principle:

\[ \boxed{\text{individually rational choices can produce a collectively worse outcome}} \]

This structure appears in problems involving:

pollution;
price competition;
arms races;
shared resources;
team contribution;
information sharing.

Coordination Games

In a coordination game, players benefit from choosing compatible actions.

Examples include:

choosing compatible technical standards;
driving on the same side of the road;
meeting at the same location;
adopting the same communication protocol.

Coordination games often have multiple equilibria.

Anti-Coordination Games

In an anti-coordination game, players benefit from choosing different actions.

Examples can include:

selecting different time slots;
avoiding the same congested resource;
specializing in different roles.

Sequential Games

Not all players move at the same time.

In a sequential game, one player may act before another.

A game tree can represent:

decision nodes;
available actions;
later responses;
terminal payoffs.

Game Trees

Suppose Player 1 chooses Enter or Stay Out.

If Player 1 enters, Player 2 chooses Fight or Accommodate.

Player 1
├── Stay Out → payoff
└── Enter
    └── Player 2
        ├── Fight → payoff
        └── Accommodate → payoff

The tree makes the order of decisions explicit.

Backward Induction

Sequential games are often solved by working backward from the final decision.

At each later decision node, identify the action that the player would prefer if that node were reached.

Then move backward to determine how earlier players should anticipate those later choices.

This is called backward induction.

Credible Threats

A threat is credible when carrying it out would actually be optimal or rational for the threatening player if the relevant point in the game were reached.

A threat that would harm the player making it may not influence a rational opponent if the opponent expects the threat will not be carried out.

Repeated Games

A repeated game is played more than once.

Repetition changes incentives because today's action can affect tomorrow's response.

Players may condition future behavior on:

past cooperation;
past defection;
reputation;
punishment;
reward.

Discounting Future Payoffs

Future payoffs may be valued less than immediate payoffs.

If \(\delta\) is a discount factor with

\[ 0<\delta<1, \]

a payoff \(u_t\) received \(t\) periods in the future may contribute

\[ \boxed{\delta^t u_t} \]

to present strategic value.

Repeated Cooperation

Repetition can make cooperation more attractive when future consequences are important enough.

A player may sacrifice a short-run gain from defection to preserve future cooperative benefits.

This depends on:

how long interaction is expected to continue;
how much future payoffs are valued;
whether past actions can be observed;
whether punishments or rewards are credible.

Incomplete Information

Players may not know everything about one another.

Unknown features may include:

costs;
preferences;
capabilities;
private information;
risk tolerance.

Games with incomplete information combine strategic reasoning with probability and Bayesian updating.

Beliefs

When another player's type or information is unknown, a player may form a probability distribution over possibilities.

Strategic decisions can then depend on expected payoff under those beliefs.

This is one way Bayesian reasoning enters game theory.

Signaling

A signal is an observable action or message that may reveal information about an otherwise hidden characteristic.

Signals are informative only when their incentives and costs make different types behave differently enough for observers to learn something.

Commitment

A commitment changes future strategic possibilities by restricting one's own later choices.

A credible commitment can influence another player's current decision.

Strategic commitment appears in:

contracts;
capacity investment;
public policies;
technical design;
negotiation.

Games with More Than Two Players

Many practical games contain multiple players.

Examples include:

traffic networks;
auctions;
political coalitions;
shared-resource problems;
market competition.

The number of possible strategy profiles can grow rapidly.

Counting Strategy Profiles

Suppose player \(i\) has \(m_i\) pure strategies.

The total number of pure-strategy profiles is

\[ \boxed{m_1m_2\cdots m_n}. \]

If four players each have three strategies, the number of profiles is

\[ 3^4=\boxed{81}. \]

Congestion Games

In a congestion game, the payoff from using a resource depends on how many other players also use it.

Examples include:

drivers choosing roads;
users choosing servers;
firms choosing shared suppliers;
customers choosing service lines.

Strategic choices can therefore create congestion externalities.

Social Optimum Versus Equilibrium

A Nash equilibrium need not maximize total welfare.

Let total welfare be

\[ W=\sum_i u_i. \]

A social optimum maximizes \(W\).

A Nash equilibrium instead requires unilateral stability.

These are different criteria.

Price of Anarchy

One way to compare decentralized strategic behavior with coordinated optimization is the price of anarchy.

Informally, it measures how inefficient an equilibrium can be relative to a socially optimal outcome.

The exact ratio depends on whether the objective is framed as a cost to minimize or a welfare measure to maximize.

Mechanism Design

Game theory can also run in reverse.

Instead of asking

what will players do under these rules?

mechanism design asks

what rules should be chosen to produce desirable incentives?

This connects strategic behavior with auctions, voting, contracts, and market design.

Common Error: Choosing the Largest Number in the Matrix

A player cannot usually choose an outcome cell directly.

Each player controls only their own strategy.

Therefore, the largest payoff anywhere in the matrix may not be attainable when the other player's incentives are considered.

Common Error: Comparing the Wrong Payoff

In a payoff pair

\[ (a,b), \]

Player 1 compares the first numbers.

Player 2 compares the second numbers.

Mixing these components produces incorrect best-response analysis.

Common Error: Calling Any Good Outcome an Equilibrium

A Nash equilibrium is not defined as the outcome with the highest total payoff.

It is defined by unilateral incentives.

At equilibrium, no player can gain by changing only their own strategy.

Common Error: Assuming Equilibrium Means Fair

An equilibrium can be unequal, inefficient, or undesirable.

Equilibrium means strategically stable under the model, not morally fair or socially optimal.

Common Error: Assuming Dominance Is Required for Nash Equilibrium

A Nash equilibrium can exist even when no player has a dominant strategy.

Dominance is a stronger condition than mutual best response.

Common Error: Averaging Pure Strategies Without Probabilities

A mixed strategy must specify probabilities.

Saying a player uses both Up and Down is incomplete unless the randomization rule is given.

For example,

\[ P(\text{Up})=0.4,\qquad P(\text{Down})=0.6. \]

Common Error: Treating Repeated and One-Shot Games as Identical

Repetition can change incentives because present actions affect future responses.

A strategy that is unattractive in a one-shot game may become sustainable when future payoffs matter.

Common Error: Ignoring Information

A player's optimal strategy can depend on what they know when choosing.

Simultaneous-move, sequential-move, complete-information, and incomplete-information games are different models.

A Reliable Procedure for Payoff-Matrix Games

Identify the players.
List each player's strategies.
Confirm the payoff order in each cell.
For each opponent strategy, identify best responses.
Check for dominant or dominated strategies.
Mark mutual best responses.
Identify pure-strategy Nash equilibria.
If none exist, determine whether mixed-strategy analysis is appropriate.
Interpret the strategic result in the original context.

A Reliable Procedure for Mixed Strategies

Assign a probability variable to one player's randomization.
Compute the opponent's expected payoff from each pure strategy.
Set those expected payoffs equal when the opponent mixes between them.
Solve for the probability.
Repeat for the other player.
Check that all probabilities lie between \(0\) and \(1\).
Calculate the equilibrium expected payoff if requested.

A Reliable Procedure for Sequential Games

Draw the game tree in decision order.
Label the player controlling each decision node.
Label each action branch.
Write terminal payoffs in consistent player order.
Start at the final decision nodes.
Select each later player's preferred action.
Work backward to earlier decisions.
Check whether threats and commitments are credible.

Example: Full Dominance and Equilibrium Workflow

	Player 2: L	Player 2: R
Player 1: U	\((2,3)\)	\((1,2)\)
Player 1: D	\((4,1)\)	\((3,4)\)

For Player 1:

\[ 4>2 \]

against L, and

\[ 3>1 \]

against R.

Therefore D strictly dominates U.

If Player 1 chooses D, Player 2 compares

\[ 1\text{ under L} \]

with

\[ 4\text{ under R}. \]

Player 2 prefers R.

Thus

\[ \boxed{(\text{D},\text{R})} \]

is a Nash equilibrium.

Example: Full Zero-Sum Workflow

Consider Player 1's payoff matrix:

\[ \begin{bmatrix} 3 & 1\\ 2 & 4 \end{bmatrix}. \]

Row minima:

\[ 1,\qquad2. \]

Therefore,

\[ \text{maximin}=\boxed{2}. \]

Column maxima:

\[ 3,\qquad4. \]

Therefore,

\[ \text{minimax}=\boxed{3}. \]

Since

\[ 2<3, \]

there is no pure saddle point.

Practice 1: Read a Payoff

A payoff cell is

\[ (5,2). \]

What payoff does Player 2 receive?

Practice 2: Best Response

If Player 2 chooses Left, Player 1 receives payoff \(3\) from Up and \(7\) from Down.

What is Player 1's best response?

Practice 3: Dominance

Player 1 receives:

\[ \text{Up}: (2,4) \]

across two opponent choices and

\[ \text{Down}: (5,6). \]

Does Down strictly dominate Up?

Practice 4: Strategy Profiles

Three players each have \(4\) pure strategies.

How many pure-strategy profiles are possible?

Practice 5: Mixed Probability

A player chooses A with probability \(0.3\).

What is the probability of choosing B if these are the only two strategies?

Practice 6: Expected Payoff

A strategy yields payoff \(8\) with probability \(0.25\) and payoff \(4\) with probability \(0.75\).

Find the expected payoff.

Practice 7: Zero-Sum Payoff

In a zero-sum game, Player 1 receives payoff \(6\).

What does Player 2 receive?

Practice 8: Discounted Payoff

A payoff of \(10\) occurs two periods in the future with discount factor

\[ \delta=0.9. \]

Find its discounted value.

Practice 9: Equilibrium Meaning

At a Nash equilibrium, can one player improve their payoff by changing only their own strategy while everyone else keeps theirs fixed?

Practice 10: Sequential Reasoning

What method is commonly used to solve a finite sequential game from its final decisions backward?

Practice Answers

Practice 1

Player 2 receives the second number:

\[ \boxed{2}. \]

Practice 2

Since

\[ 7>3, \]

Player 1's best response is

\[ \boxed{\text{Down}}. \]

Practice 3

Yes.

Down gives

\[ 5>2 \]

in the first case and

\[ 6>4 \]

in the second.

Therefore, Down strictly dominates Up.

Practice 4

\[ 4^3=\boxed{64}. \]

Practice 5

\[ 1-0.3=\boxed{0.7}. \]

Practice 6

\[ (0.25)(8)+(0.75)(4) = 2+3 = \boxed{5}. \]

Practice 7

Player 2 receives

\[ \boxed{-6}. \]

Practice 8

\[ \delta^2(10) = (0.9)^2(10) = 0.81(10) = \boxed{8.1}. \]

Practice 9

No.

At a Nash equilibrium, no player can improve by changing only their own strategy while the others remain fixed.

Practice 10

The method is

\[ \boxed{\text{backward induction}}. \]

What to Remember

Games and strategic behavior arise when one decision maker's outcome depends on the decisions of others.

players make strategic choices;
payoffs represent preferences over outcomes;
best responses depend on opponents' strategies;
dominant strategies outperform alternatives across opponent choices;
Nash equilibrium consists of mutual best responses;
some games require mixed strategies;
zero-sum games align one player's gain with another's loss;
sequential games are often analyzed by backward induction;
repetition can change incentives;
incomplete information introduces beliefs and probability;
equilibrium need not be socially optimal or fair.

The central habit is: do not ask only what action gives the highest payoff; ask what the other players are likely to do and how each player's best response changes with those choices.

Where We Go Next

Game theory models decisions that depend on the behavior of other decision makers. The next applied problem family uses stochastic processes to model systems whose populations change over time through birth, death, reproduction, transition, infection, migration, or survival.

The next lesson, Biological and Population Processes, develops discrete and stochastic models for population growth, branching, survival, transitions, and biological change.