Impatience, Self-control and Strategic Thinking

Discounting the Future

Discounting the future captures the human tendency to value immediate rewards more than delayed ones — a core idea in intertemporal choice, the trade-off between now and later.

The Marshmallow Experiment: Patience Measured

Walter Mischel (Stanford) placed children in a room with one marshmallow and gave a choice: eat it now, or wait 10 minutes and get a second marshmallow. The child could ring a bell to summon the experimenter and eat the single marshmallow at any time. Key results:

• Reward in sight → average wait time ≈ 1 minute
• Reward out of sight → average wait time ≈ 11 minutes

Follow-up studies (~10 years later, 500 children traced, one-third re-interviewed) found that the waiting time was a significant predictor of later-life outcomes: SAT scores, career success, health, income. In essence, the experiment measured patience as a stable trait.

Intertemporal choice is the foundational idea: all everyday decisions involving time trade-offs — savings, education, health, exercise, entertainment — require weighing present and future utility.

Time Preference and the Discount Factor δ

From today’s perspective, consumption today is worth more than the same consumption tomorrow. The discount factor $\delta$ captures this devaluation:

$$\delta = \text{rate at which future utility is discounted}, \quad 0 < \delta < 1$$

• $\delta$ near 1: patient (future valued almost as highly as present)
• $\delta$ near 0: impatient (future nearly worthless)

Example: Total utility from dinner today ($u_0$) and dinner next week ($u_1$) is

$$U_0 = u_0 + \delta_w \, u_1$$

where $\delta_w$ is the weekly discount rate.

Intuitive classification

Exponential Discounting: The Standard Model

The standard economic (classical) model of intertemporal choice, due to Paul Samuelson, assumes exponential discounting. The present value at time 0 of a stream of future utilities is

$$U_0 = u_0 + \delta u_1 + \delta^2 u_2 + \delta^3 u_3 + \cdots = \sum_{t=0}^\infty \delta^t u_t$$

Worked Example 1: Choosing among alternatives

Given $\delta = 0.9$ and four alternatives:

Compute present values:

• $U_0(A) = 1$
• $U_0(B) = 0 + 0.9 \times 3 = 2.7$
• $U_0(C) = 0 + 0 + 0.9^2 \times 4 = 0.81 \times 4 = 3.24$
• $U_0(D) = 10$

Choice: among A,B,C → C; among all → D.

If $\delta$ drops to $0.1$:

• $U_0(A)=1$
• $U_0(B)=0.1 \times 3 = 0.3$
• $U_0(C)=0.01 \times 4 = 0.04$
• $U_0(D)=10$

Choice: among A,B,C → A; among all → D. Lower $\delta$ moves preference toward immediate consumption.

Worked Example 2: Finding δ from indifference

Ali has utility $u(x) = \sqrt{x}$. He is indifferent between:

• A: 100 today
• B: 121 one year from now

Set $U_0(A) = U_0(B)$:

$$\sqrt{100} = \delta \sqrt{121} \quad\Rightarrow\quad 10 = \delta \times 11 \quad\Rightarrow\quad \delta = \frac{10}{11} \approx 0.909$$

(A separate example uses $\delta = 2/3$; the worked example above applies the same logic to its own utility function and inputs.)

Dynamic Time Consistency of Exponential Discounting

Dynamic time consistency means that choices between two options do not change simply because time passes. A person using exponential discounting (an “econ”) has future selves that agree with the current self’s plan.

Example (IPL tickets):

• Option A: League match this year (utility 100)
• Option B: Semi-final next year (utility 150)
• Option C: Final two years from now (utility 180)
• Discount rate: 10% per year → $\delta = 0.9$

At time 0:

• $U_0(C) = 0.9^2 \times 180 = 0.81 \times 180 = 146$
• $U_0(B) = 0.9 \times 150 = 135$
• $U_0(A) = 100$ → Ali chooses C.

At time 1 (now one year later, only B and C remain):

• $U_1(C) = 0.9 \times 180 = 162$
• $U_1(B) = 150$ (since it’s consumed today) → Ali still chooses C. The plan is time consistent.

Anomalies: Dynamic Time Inconsistency for Humans

Real humans often exhibit dynamic time inconsistency: optimal plans change as time passes, all else equal. The present self and future self disagree.

• Today’s choice: wake up early tomorrow and exercise or sleep in?
• Choice made today for next week: same question — but the answer often differs.

Common statements betray inconsistency:

“I will watch IPL today, but I will complete the assignment tomorrow.” “I will go out this evening, but tomorrow I will definitely go to the gym.” “Next month I am quitting smoking.”

Plans to do the “good thing” in the future are rarely followed when that future arrives. Humans seek instant gratification now, but plan to be patient later.

Exam tip: The key distinction is that an econ (exponential discounter) is time-consistent; a human is time-inconsistent. Any question about “why do people make plans they don’t keep?” points to this anomaly.

Key takeaways

• Discount factor $\delta$ (0 < δ < 1): high = patient, low = impatient.
• Exponential discounting yields $U_0 = \sum \delta^t u_t$ and ensures dynamic time consistency — choices stable over time.
• Anomaly: humans show time inconsistency — early selves plan patience, later selves demand instant gratification.
• Real-life intertemporal choices (savings, health, education, addiction) all hinge on δ and on whether discounting is exponential or hyperbolic (implied by inconsistency).

Present-Biased Discounting

Present-biased discounting (also called hyperbolic discounting or beta-delta discounting) captures the fact that humans are much more impatient when trade-offs involve the immediate present versus the near future than when both outcomes lie further ahead. While an econ (a rational, exponential discounter) has a constant discount rate, a human exhibits a higher discount rate in the short run and a lower one in the long run.

Formal Model: Beta-Delta Discounting

The utility function evaluated at time $0$ (today) for a stream of future utilities is:

$U_0 = u_0 + \beta\delta u_1 + \beta\delta^2 u_2 + \beta\delta^3 u_3 + \dots$

Equivalently, factoring out $\beta$:

$U_0 = u_0 + \beta\left\{\delta u_1 + \delta^2 u_2 + \delta^3 u_3 + \dots\right\}$

• $\beta$ (the “present-bias” parameter) captures the extra impatience between the present and the immediate future.
• $\delta$ (the “long-run discount factor”) captures how patient the individual is when both outcomes are in the future.

Key insight: $\beta$ is typically much smaller than $1$ (e.g., $\beta = 3/5$), while $\delta$ is very close to $1$ (e.g., $\delta \approx 1$). This means the bulk of impatience is concentrated on the first step – the distance between “now” and “soon”. In long-run trade-offs (e.g., year 365 vs year 366) the individual is almost perfectly patient.

Worked Example: Rani’s IPL Tickets

Rani has three options:

Rani is present‑biased. Her discounting is:

• Immediate future (0→1): 30% discount rate → discount factor $0.7$
• Next year (1→2): 10% discount rate → discount factor $0.9$
• Thereafter: 0% → discount factor $1$

Planning at time 0

$$\begin{aligned} V_0(A) &= 100 \\ V_0(B) &= 0.7 \times 150 = 105 \\ V_0(C) &= 0.7 \times 0.9 \times 180 = 0.63 \times 180 = 113 \end{aligned}$$

Rani plans to watch the Final in year 2 (highest value).

At time 1 (when the semi‑final is available now)

$$\begin{aligned} V_1(B) &= 150 \quad\text{(no discounting – immediate)} \\ V_1(C) &= 0.7 \times 180 = 126 \end{aligned}$$

Now Rani chooses the Semi‑final. Her choice is time‑inconsistent: what she planned at time 0 (Final) is reversed when the moment arrives.

Empirical Evidence

McClure et al. (thirsty subjects)

Estimated discount rates: short‑run ≈ 50%, long‑run ≈ 0%.

Thaler (early present‑bias study)

The huge gap in implied rates (345 % vs 19 %) is direct evidence that short‑run impatience far exceeds long‑run impatience.

Preference Reversals

Preference reversal is the signature of present bias: people systematically change their choice when the outcome moves from “future” to “now”.

Snack choice (Read & van Leeuwen)

• Plan for next week: 74% choose apple (healthy).
• Choose for today: 70% choose fries (unhealthy).

Movie choice (Lowenstein et al.)

• Tonight: 66% choose a low‑brow movie (e.g., Fast & Furious).
• Same day next week: 37% choose low‑brow (majority choose high‑brow, e.g., The Seventh Seal).
• Two weeks from now: only 29% choose low‑brow.

Applications

Credit card vs Personal loan

A consumer wants a ₹70,000 TV.

• Credit card: 20 % p.a. (easily swiped now).
• Personal loan: 8 % p.a. (requires going to a bank, paperwork).

Present‑biased consumers swipe the card despite the higher interest, because the immediate cost of effort looms larger than the delayed interest saving.

Exam tip: A salesperson knows this – they will try to make the purchase immediate and the payment distant, exploiting the customer’s present bias.

Vices and Virtues

Present bias explains why we systematically prepone vices and postpone virtues.

Formally, let:

• $\beta = \frac12$, $\delta = 1$ (all impatience is in the first step).
• Benefit $B$ and cost $C$ occur at different times.

Vices: Immediate benefit, delayed cost

Example: an egg roll – benefit 3 (taste now), cost 4 (health later).

Result: preponement – the vice is consumed despite the plan to abstain.

Virtues: Immediate cost, delayed benefit

Example: exercise – benefit 10 (future health), cost 8 (effort now).

Result: postponement – the virtue is never done despite good intentions.

Preference reversal for vices and virtues arises from the same mechanism: the immediate component (benefit for vices, cost for virtues) is not discounted, while the delayed component is discounted by $\beta$.

Key takeaways

• Present‑bias (hyperbolic/β-δ) discounting: $\beta$ short‑run impatience, $\delta$ long‑run patience; $\beta < 1$, $\delta \approx 1$.
• Utility at time 0: $U_0 = u_0 + \beta\sum_{t\ge1}\delta^t u_t$.
• Leads to time‑inconsistent plans: what you plan today differs from what you actually choose later.
• Preference reversals are pervasive: snacks, movies, credit decisions.
• Vices (immediate benefit, delayed cost) are preponed; virtues (immediate cost, delayed benefit) are postponed – both explained by the same β-δ model.

Exam tip: The classic preference‑reversal problem asks: “At time 0 Rani plans the Final, at time 1 she picks the Semi‑final.” Be ready to compute the discounted values and show the reversal exactly as in the worked example.

The Self-Control Problem: A Motivation

At a dinner party, Richard Thaler removed a bowl of cashews because he could not stop eating them. His economist colleague objected: if you prefer not to eat more, why not just stop? Thaler’s reply — “if the nuts were still available, I would have eaten more” — reveals a self-control problem that standard economic theory cannot capture. The “human” experiences a conflict between present passion and future preferences; the “econ” assumes preferences are consistent across time.

Further examples:

• Smokers buy single sticks to avoid chain-smoking a pack.
• Dieters avoid stocking ice cream at home.
• People place alarm clocks out of reach.
• Academics register for conferences far in advance to force paper completion.
• Buying a smaller, more expensive Coke to limit consumption.

Adam Smith’s Theory of Moral Sentiments described this as a conflict between passion and the impartial spectator (reason). He noted that pleasures ten years hence matter little compared to today’s, and that willpower is needed to overcome myopia.

Exponential vs. Hyperbolic (Present‑Bias) Discounting

Under exponential discounting, if you do not indulge now, you will not indulge in the future — preferences are time‑consistent. Under hyperbolic (present‑bias) discounting, if you do not indulge now, you may want to indulge later — preferences can reverse over time. This creates the distinction between naifs and sophisticates.

Worked Example: Movie Theatre Scheduling

You can watch a movie on three of four weekends. Each weekend offers a different film:

You must skip exactly one movie. Assume $\beta = \frac12$, $\delta = 1$ (i.e., no long‑run discounting; all present bias is immediate).

Exponential Discounter (β irrelevant)

At time 0, evaluate utilities of skipping each movie:

• Skip A: $3 + 5 + 8 + 13 - 3 = 5 + 8 + 13 = 26$
• Skip B: $3 + 8 + 13 = 24$
• Skip C: $3 + 5 + 13 = 21$
• Skip D: $3 + 5 + 8 = 16$

Best is to skip the worst movie (A). Choice is time‑consistent.

Naive Hyperbolic Discounter

At time 0, the current period is weekend 0. Any utility received in future periods ($t \ge 1$) is discounted by $\beta = \frac12$.

$$\text{Utility(time 0) of skipping }X = \begin{cases} \text{(current utility)} + \frac12 \times (\text{future utilities}) & \text{if } X \text{ is not in current period}\\ \frac12 \times (\text{all other utilities}) & \text{if } X \text{ is in current period} \end{cases}$$

Calculations at time 0:

• Skip A (current): $\frac12 (5 + 8 + 13) = 13$
• Skip B (future): $3 + \frac12(8 + 13) = 3 + 10.5 = 13.5$
• Skip C (future): $3 + \frac12(5 + 13) = 12$.
• Skip D (future): $3 + \frac12(5 + 8) = 9.5$.

So at time 0, highest utility is 13.5 → plans to skip B.

At time 1 (now current period is B, utility 5):

• Skip B: $\frac12(8+13) = 10.5$
• Skip C: $5 + \frac12(13) = 5 + 6.5 = 11.5$
• Skip D: $5 + \frac12(8) = 5 + 4 = 9$

Highest = 11.5 → skips C (not B as planned).

At time 2 (current period C, utility 8):

• Skip C: $\frac12(13) = 6.5$
• Skip D: $8$.

The higher utility is 8, so the individual skips D (the fantastic movie).

Final outcome: Naif ends up skipping the best movie (D) — a clear failure to follow the original plan.

Sophisticated Hyperbolic Discounter

The sophisticate knows at time 0 that her future self will behave according to present bias. She solves the problem by backward induction (subgame perfect Nash equilibrium).

At time 2 (if reached with movies C and D left):

• Skip C: $\frac12(13) = 6.5$
• Skip D: $8$ So she will skip D (watch C). Thus, at time 2, C is definitely watched.

At time 1 (movies B, C, D left, but knows C will be watched at time 2): Effective choices: skip B or skip D (since C is already consumed later).

• Skip B: $\frac12(8+13) = 10.5$
• Skip D: $5 + \frac12(8) = 9$ Because $10.5 > 9$, she skips B at time 1. The resulting set of watched movies is A, C, and D.

At time 0 (movies A, B, C, D; but she knows future decisions): Anticipating the later choices, she compares the two feasible initial plans: skip A (utility 13) or skip B (utility 13.5). She chooses to skip B, and the time-1 decision confirms that plan. The final set is A, C, and D.

Key insight: Naifs are time‑inconsistent (plan changes); sophisticates are time‑consistent (plan holds) because they anticipate future self‑control failures and incorporate them into the initial decision.

Key Takeaways

• Naifs believe future selves share current preferences ($\hat{\beta}=1$); they plan optimally but re‑optimise later, leading to time‑inconsistent choices (end up skipping the best option).
• Sophisticates know their future selves will be present‑biased ($\hat{\beta}=\beta$); they solve a dynamic game with future selves and make a time‑consistent plan (pre‑commit indirectly).
• Partial naifs have $\beta < \hat{\beta} < 1$ and display intermediate behaviour.
• The core distinction is not about having a self‑control problem, but about awareness of it.
• Exam tip: In any intertemporal choice problem with present bias, first identify whether the decision‑maker is naive or sophisticated. The naive always revise plans; the sophisticated use backward induction to find a stable plan.

Commitment Contracts

Sophisticated individuals—those who are aware of their own self-control problems—take deliberate steps to avoid falling prey to temptation. The core idea: if you know you will give in later, you can constrain your future self now. Two general strategies emerge:

1. Remove cues – eliminate the triggers that prompt temptation.
2. Limit one’s own choices – make it physically or financially costly to give in.

These are implemented through commitment devices (or commitment contracts): arrangements that raise the cost of yielding or block the tempting option entirely.

Ulysses and the Sirens (The Classic Example)

Ulysses knew that if he heard the irresistible songs of the Sirens, he would steer his ship onto the rocks and die. He wanted to hear the music but survive.

Result: he enjoyed the songs and lived. This story illustrates the two strategies: eliminating temptation cues (for the crew) and pre‑committing to a constrained choice (for himself).

Modern Commitment Devices

A set of products that operationalise the same logic:

Exam tip: Commitment devices work by making indulgence more expensive than abstaining, thereby shifting the relative cost of giving in.

Field Evidence on Commitment Contracts

1. CARES – Smoking Cessation (Gine, Karlan, Zinman)

• Programme: smokers deposit funds into a savings account for six months. After six months, they take a nicotine and cotinine test.
  • Pass → money returned.
  • Fail → money forfeited.
• Results:
  • Take‑up: ~11% of those offered.
  • Those offered CARES were 3 percentage points more likely to pass the test at 6 months.
  • Effects persisted in a surprise test at 12 months.
• Interpretation: effects exist but are modest; only a self‑selected sophisticated minority signs up.

2. Factory Workers – “Dominated” Commitment Contract (Kremer et al.)

Setting: workers produce baskets at a piece rate.

• Blue schedule: constant wage $w$ per unit.
• Red schedule:
  • If production $< T$: wage $w/2$ per unit.
  • If production $\geq T$: wage $w$ per unit (same as blue on those units).

The red schedule is dominated – it pays no more than blue for any output level, and strictly less if production falls short of $T$. A rational agent (without self‑control problems) would never choose it.

Yet 35% of workers chose red.

Why? Sophisticated workers who know they will slack off just below $T$ use the red contract as a goal‑setting device. The threat of earning only $w/2$ pushes them to cross $T$, so total earnings actually increase.

Key: the factory owners did not raise wages, yet workers produced more and earned more – a Pareto improvement from a commitment contract.

3. Gym Attendance (Royer, Stehr, Sydnor)

• Participants could commit to visit the gym at least once every 14 days.
• They chose how much money to put at stake; failure → money donated to charity.
• Results:
  • Full sample take‑up: 13%.
  • Among existing gym members: 25%.
  • Among non‑members: 6%.
  • Average amount committed: $63 (non‑trivial).

Take‑up is again low, but those who commit stake real money.

Exam tip: Across all studies, take‑up of commitment contracts is modest (≈10–35%). Sophistication is not universal – many people are naive and do not realise they need help. Commitment contracts work for the sophisticated minority who opt in.

Key takeaways

• Sophisticated agents use commitment devices to fight self‑control problems: remove cues or limit choices.
• Lab and field examples (Ulysses, Shreddy, stickk) show the same logic.
• Dominated contracts (red schedule) can be chosen to create external goals.
• Field evidence: CARES, factory contracts, gym commitment all show positive but modest effects, with low take‑up.
• Commitment contracts work best for those who recognise their own weakness.

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Patience and the Wealth of Nations

The relationship between patience (willingness to trade current consumption for future benefit) and national economic outcomes is studied using cross‑country data (2018 paper covering 76 countries: 15 Americas, 20 Europe, 22 Asia, 14 Africa).

Measure of patience: a combination of quantitative and Likert‑scale questions (e.g., “How willing are you to give up something today for tomorrow’s benefit?” – consistently measured across countries).

Key Correlations

Patience is not just an individual trait; it predicts major macroeconomic differences. More patient countries tend to be richer, grow faster, save more, and educate longer.

Exam tip: This is often tested as a “big picture” takeaway: patience correlates with development, but causality is complex (reverse causality, institutions, etc.). Treat this as correlation, not causation.

Key takeaways

• Cross‑country data shows patience is positively correlated with GDP per capita, growth, savings, and schooling.
• The global distribution of patience varies systematically with development.
• Patience matters not only for individual inter‑temporal choice but for national outcomes.

Strategic Thinking among Econs: Game Theory

Strategic interaction is modelled using game theory. The standard framework assumes rational, self-interested players who think through the consequences of their choices—and the choices of others—to arrive at an equilibrium.

The Beauty Contest Game (Motivation)

Before formalising strategic thinking among econs, consider a simple game that reveals the depth of reasoning required:

• Each player guesses a number between 0 and 100.
• The target is $\frac{2}{3}$ of the average of all guesses.
• The player whose guess is closest to the target wins.

A rational player reasons iteratively: if everyone picks 50, the target is 33; if everyone picks 33, the target is 22; iterating leads to 0. This illustrates the logic of common knowledge of rationality—a key assumption in game theory.

Classification of Games

Games are broadly divided into non-cooperative (players cannot form binding agreements) and cooperative (binding agreements allowed). Non‑cooperative games further split into:

• Simultaneous move: All players choose actions at the same time, each ignorant of the others’ choices.
• Sequential move: Players move in a known order; later players observe earlier moves.

Ingredients of a Strategic Environment

Key Assumptions

• Rationality: Each player chooses the action that maximises her own payoff, and forms correct beliefs about the world.
• Common knowledge: (i) Every player is rational. (ii) Every player knows that every player is rational. (iii) Every player knows that (ii) holds, ad infinitum. This iterative reasoning drives many game‑theoretic predictions.

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Prisoner’s Dilemma

Developed at RAND in 1950, this game models a situation where cooperation is collectively beneficial but individually dominated by defection.

Story (from Dixit & Nalebuff): Two suspects are arrested. If both remain silent, each gets 1 year. If one implicates the other (confesses) while the other stays silent, the confessor goes free and the silent one gets 10 years. If both confess, each gets 5 years.

Payoff matrix (years in jail; lower is better):

Dominant strategy – a strategy that is best regardless of what the other player does.

• For Prisoner 1: −5 > −10 (if 2 confesses) and 0 > −1 (if 2 does not) → Confess is strictly better.
• By symmetry, Confess is also dominant for Prisoner 2.

Thus (Confess, Confess) is the dominant‑strategy equilibrium. The dilemma: both would prefer (−1, −1) but rational self‑interest leads to (−5, −5).

Exam tip: The Prisoner’s Dilemma is the classic example of a game where the dominant strategy yields a Pareto‑inferior outcome. It explains why cooperation may fail even when it benefits all.

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Stag Hunt & Nash Equilibrium

Not every game has a dominant strategy. The Stag Hunt game illustrates:

• Two hunters independently choose to hunt Stag (requires cooperation) or Hare (safe but smaller reward).
• If both hunt Stag, they get a large payoff (5 each). If one hunts Stag alone, they get 0. Hunting Hare alone yields 2. Both hunting Hare yields 2 each.

Payoff matrix:

• No dominant strategy: Player 1’s best response depends on Player 2’s choice.
• Nash equilibrium (John Nash): A set of strategies where no player can improve her payoff by unilaterally changing her own strategy, given the other player’s strategy.
• The Stag Hunt has two Nash equilibria: (Stag, Stag) and (Hare, Hare). Neither player wants to deviate alone from either outcome.

Nash proved that every finite game has at least one Nash equilibrium (possibly in mixed strategies).

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Extensive‑Form Games & Subgame Perfect Nash Equilibrium

When moves are sequential, the extensive form (game tree) captures the order of play, decision nodes, and payoffs. Additional reasoning allows refining predictions beyond Nash equilibrium.

Example :

• Player 1 chooses A or B.
• If A, game ends: payoffs (1 for P1, 2 for P2).
• If B, Player 2 chooses L or R.
  • L → (0, 0)
  • R → (2, 1)

Normal‑form representation – Player 2’s strategies are contingent on Player 1’s move: BL (if P1 plays B, then L) and BR (if P1 plays B, then R). Player 1 has strategies A and B.

• Nash equilibria: (A, BL) and (B, BR). Both are self‑enforcing in the normal form.

Subgame Perfect Nash Equilibrium (SPNE) – A refinement that requires the strategies to be a Nash equilibrium in every subgame (every decision node). It rules out non‑credible threats.

• Analyse the last node: Player 2 will choose R (1 > 0). Player 1, foreseeing this, compares 1 (from A) with 2 (from B, followed by R), and chooses B.
• SPNE: Player 1 plays B, Player 2 plays R → (B, BR). This eliminates the (A, BL) equilibrium, which relied on Player 2 threatening to play L if B were chosen (a non‑credible threat).

Key takeaway: In sequential games, backward induction (starting from the last decision) yields the SPNE, providing a sharper prediction than Nash equilibrium alone.

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Exam tip: The Stag Hunt’s two Nash equilibria illustrate coordination problems. The subgame perfect equilibrium (SPNE) is the standard solution concept for sequential games—always check for non‑credible threats.

Key takeaways

• Game theory models strategic interaction among rational players. Key assumptions: rationality and common knowledge.
• Games can be simultaneous or sequential, one‑shot or repeated.
• Dominant strategy – best regardless of others; leads to a dominant‑strategy equilibrium (e.g., Prisoner’s Dilemma).
• Nash equilibrium – no player can gain by unilaterally deviating; every finite game has at least one.
• Subgame Perfect Nash Equilibrium refines Nash equilibrium for sequential games by requiring credibility at every decision node.
• The Prisoner’s Dilemma and Stag Hunt are canonical models: one illustrates the failure of cooperation under self‑interest; the other illustrates multiple equilibria and the need for coordination.

Strategic Thinking Among Humans: Level K Thinking

Level K thinking models bounded rationality: a player assumes others are one step less sophisticated and chooses the best response to that belief. The Beauty Contest Game and the Centipede Game reveal how people differ in their depth of reasoning—and why playing the rational equilibrium is often not optimal unless opponents are also rational.

The Beauty Contest Game

John Maynard Keynes introduced the beauty contest analogy to explain stock‑price volatility. In a newspaper contest, readers pick the six most attractive faces; winners are those whose choices match the most popular selection. Keynes wrote that a smart player does not choose the prettiest face, nor even the average opinion of prettiest, but “anticipates what average opinion expects the average opinion to be”—third‑degree thinking, with higher degrees possible.

Numerical version: Guess a number from 0 to 100. The winner is the player whose guess is closest to $\frac{2}{3}$ of the average of all guesses.

Iterative elimination of dominated strategies

Start from the worst case: if everyone guessed 100, the dominated strategy would be any number above $\frac{2}{3}\times100 = 66\frac{2}{3}$. Thus the effective range shrinks to $[0, 66\frac{2}{3}]$.

Repeating the logic:

• From $[0, 66\frac{2}{3}]$: dominated region > $\frac{2}{3}\times66\frac{2}{3} = 44\frac{4}{9}$ → new range $[0, 44\frac{4}{9}]$.
• Next iteration: dominated region > $\frac{2}{3}\times44\frac{4}{9} = 29\frac{17}{27}$ → range shrinks further.

With enough iterations, every number except 0 is dominated. The Nash equilibrium (and also a dominant‑strategy equilibrium) is to guess 0.

Level‑K reasoning in practice

Actual players rarely reach 0. Instead they exhibit Level K thinking:

Exam tip: Only an infinite‑level thinker plays the Nash equilibrium. Real data show masses at 33 (Level‑1) and 22 (Level‑2), not at 0.

Empirical evidence (Bosch‑Domènech, Nagel & Satorra):

• Lab and classroom experiments: strong peaks at guesses 33 and 22.
• Theorists (economists playing the game): high mass at 0, i.e., they play the Nash equilibrium.

Key takeaways – Beauty Contest

• The game illustrates iterated elimination of dominated strategies → equilibrium = 0.
• Players display finite levels of strategic thinking (Level‑0, Level‑1, … ).
• Most people are Level‑1 or Level‑2; only trained theorists frequently reach the Nash equilibrium.

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The Centipede Game

A two‑player sequential game with six decision nodes. White moves first; each player can Stop or Continue. Payoffs (White, Black) at each terminal node:

• White stops at first node: (4, 1)
• Black stops at second node: (2, 8)
• White stops at third node: (16, 4)
• Black stops at fourth node: (8, 32)
• White stops at fifth node: (64, 16)
• Both continue to final node: (256, 64)

Subgame Perfect Nash Equilibrium (SPNE): By backward induction, each player stops as soon as they have the move → White stops at the first node, Black never gets to play.

Do players play the SPNE?

Experimental evidence (using chess players of different ranks and college students) shows that who you play against determines whether the SPNE is chosen.

Stopping proportions by node (data from the study):

Note: Percentages are approximate; nodes are labelled from first (1) to sixth (6).

Key patterns:

• Grandmasters stop at the very first node most often (69% when playing another chess player).
• College students rarely stop early; they continue further into the game (especially when playing other students).
• Students adapt to opponents: when playing against a chess player, they stop much earlier (28% at node 1, 36% at node 2) than when playing another student.
• Chess players also adapt: they stop at node 1 only 37% of the time when facing a student, but 69% when facing another chess player.

Core insight: The rational outcome (SPNE) is played only when both players are highly rational and believe the opponent is similarly rational. If a player believes the opponent is naive (e.g., a student who will continue), it becomes optimal not to stop at the first node.

Key takeaways – Centipede

• SPNE calls for immediate stop, but actual behavior depends heavily on opponent type.
• Sophistication is relative: grandmasters vs. students show clear differences in stopping depth.
• Rationality is not absolute—players adjust their level of strategic thinking based on who they face.

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Overall Key Takeaways for Level K Strategic Thinking

• Beauty contest: Nash equilibrium (0) is achieved only by infinite‑level thinkers; real players cluster at Level‑1 (33) and Level‑2 (22).
• Centipede game: SPNE (stop immediately) is rarely played; experienced chess players approach it, but only when matched against each other.
• Opponent matters: how far you “think ahead” is endogenous—if you know your opponent is unsophisticated, it can be rational to deviate from the equilibrium.
• The Level K model provides a tractable way to describe bounded rationality and predict behavior in strategic settings.

Cognitive Hierarchy Model

The Cognitive Hierarchy Model (CHM), introduced by Camerer, Ho & Chong, generalises level‑k thinking by assuming players differ in how deeply they reason about others’ choices. Instead of a single step level, each player has a type $k$ that reflects how many steps of strategic reasoning they perform.

• Type 0 – does not consider competitors’ choices (naive, random or non‑strategic).
• Type 1 – believes all others are Type 0.
• Type 2 – believes others are a mix of Type 0 and Type 1, according to some distribution.
• Type k – believes others are distributed among Types 0 through $k-1$ with a given distribution.

The distribution of types across a population is modelled by a Poisson distribution with a single parameter $\tau$ (tau). The probability of a randomly chosen player being Type $k$ is:

$P(\text{Type }k) = \frac{e^{-\tau} \tau^k}{k!}$

As $\tau$ increases, more players are of higher types (more sophisticated strategic thinking).

Key intuition: $\tau$ measures the average depth of reasoning in the population. Higher $\tau$ → more players think multiple steps ahead.

Worked illustration

For $\tau = 2$:

• Type 0 ≈ 13%
• Type 1 ≈ 27% (draw vertical at $k=1$, ~22% in original figure; values vary by rounding)
• Type 2 ≈ 27% (~29% in figure)
• Type 3 ≈ 18%

The exact percentages come from the Poisson formula.

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Application: ISP Market Entry (Goldfarb & Yang, “Are All Managers Created Equal?”)

Context: 1997 US ISP market – firms chose whether to adopt 56K modem technology in a short, three‑month window. This is a simultaneous‑move game with heterogeneous managers.

Key questions:

1. Does strategic thinking (higher $\tau$) affect technology adoption?
2. Do firms with higher $\tau$ survive longer?
3. What characteristics correlate with $\tau$?

Estimating $\tau$ for each firm

Researchers estimated:

$\log(\tau^j) = \gamma_0 + \gamma_1 z_{ij}$

where $j$ indexes ISP, $i$ indexes market, and $z$ includes:

• Market‑level: number of competitors, % urban population, % with college degree
• Firm‑level: number of markets served

Two specifications:

1. Full model with all covariates.
2. Restricted model: $\log(\tau) = \gamma_0$ (same $\tau$ for all firms).

Results

Findings:

• Firms in markets with more educated populations, more competitors, and urban areas have higher $\tau$ → more strategic managers.
• Firms with higher $\tau$ were more likely to survive through April 2007 (via continued operation or acquisition).
• Higher $\tau$ also correlated with higher profits.
• The study reports correlations, not causal effects.

Takeaway

The degree of strategic sophistication ($\tau$) matters for business performance. A Nash equilibrium assumption that all players are fully rational may be inferior to accounting for competitors’ actual reasoning depth.

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Key takeaways – Cognitive Hierarchy Model

• Players are distributed by type $k$ (0,1,2,…) following a Poisson($\tau$) distribution.
• Type 0 is non‑strategic; higher types reason about lower types.
• $\tau$ captures average depth of thinking; higher $\tau$ = more sophisticated.
• In the ISP study, $\tau \approx 2.66$ on average. Firms with higher $\tau$ correlated with higher survival and profits.
• Effective strategic choice requires knowing (or estimating) $\tau$ in your market.

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Idea of Fairness – I

Fairness perceptions often conflict with standard economic predictions (supply‑demand, profit maximisation).

Fairness experiments & thought experiments

1. Monkey fairness (Frans de Waal): Capuchin monkeys perform a simple task (give a stone) and receive a reward. If one monkey gets cucumber while the other gets grapes (preferred), the under‑rewarded monkey shows distress – a basic sense of inequity.

2. Salary disparity: You are offered Rs. 20 lakhs, non‑negotiable, but discover a batchmate with similar qualifications received Rs. 25 lakhs from the same company. Would your effort change? Many report lower effort – perceived unfairness reduces motivation.

3. Price gouging during disaster: After Cyclone Amphan, a store raises bottled water to Rs. 100. Most judge this as “very unfair” (‑2), even though standard economics says prices should rise with excess demand.

4. Snow shovels (Kahneman, Knetsch, Thaler): A hardware store sells shovels for $15. After a snowstorm, it raises the price to$20. 82% of survey respondents said “unfair.” In Thaler’s MBA class at UChicago, >76% said “unfair” – despite having learned that price increases during shortages are efficient.

5. Cabbage patch dolls at Christmas: A store discovers one doll and announces an auction to the highest bidder. 74% consider this unfair. Reasons: the doll goes only to an affluent child; store exploits desperate parents.

   • Variation: If auction proceeds go to UNICEF, 79% consider the same auction “fair” or “very fair.” The purpose of the profit changes fairness perception.

Why fairness matters for business

Firms must anticipate that consumers’ fairness judgments affect willingness to buy, effort, and loyalty. Framing is critical.

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Additional examples

• Flu medicine auction: A small town has one package left; auctioning it is widely seen as unfair.
• Kidney purchase: A rich person buys a kidney from a poor person – perceived as exploitation, not a market transaction.

These all involve excess demand and high willingness‑to‑pay, yet people reject the market outcome as unfair.

Framing and the endowment effect

The same economic change can be seen as fair or unfair depending on how it is presented relative to the status quo.

Automobile dealer example:

• Scenario A: Car usually sold at list price. Shortage → dealer adds $200 surcharge (new price = list +$200). 71% say unfair.
• Scenario B: Car usually sold at $200 below list price (a discount). Shortage → dealer removes discount, sells at list price. 58% say fair.

The net price is identical in both scenarios ($200 above the original list in A, list price in B), but because scenario B frames the change as removing a discount (staying within the original list price), it is seen as far less objectionable.

Strategic implication: Set the highest intended price as the regular price. Any later price reduction can be framed as a “sale”; removing a discount is less objectionable than adding a surcharge – even when the monetary effect is the same.

Norms and culture

Fairness expectations depend on status quo, which is shaped by culture.

• In Italy, charging extra for eating in vs. takeaway is normal; in the US it seems unfair.
• Tipping is expected in the US but not in much of Europe; not tipping is seen as unfair in the US.

Key takeaways – Idea of Fairness

• People’s fairness judgments often contradict standard supply‑demand logic.
• Price increases during shortages (gouging) or auctioning essentials are widely condemned.
• The purpose of the profit matters: donating proceeds to charity can restore fairness.
• Framing matters: removing a discount is perceived as more fair than adding a surcharge, even if price is the same.
• Status quo and cultural norms anchor what is considered fair.
• Managers should account for fairness perceptions in pricing and strategy – they affect real outcomes (effort, loyalty, reputation).

Downward Nominal Wage Rigidity

When the economy booms, wages rise; during recessions, wages do not fall — or fall too little to clear the labour market. This phenomenon is downward nominal wage rigidity, a core building block of Neo-Keynesian macroeconomics. Intuitively: firms prefer to fire workers rather than cut everyone’s pay because wage cuts enrage employees and destroy productivity.

Supreet Kaur’s influential work shows that nominal wages in India are downward rigid: even when unemployment is high, employers cannot reduce wages without triggering retaliation.

Manager’s dilemma when the firm is underperforming:

Exam tip: Downward wage rigidity explains involuntary unemployment during recessions — wages do not fall enough to restore full employment.

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Fairness and Wage Cuts: Evidence

Experiments reveal that people judge fairness based on nominal rather than real wages.

The key: workers react to changes in the nominal wage, not the real wage. A 5% raise with high inflation feels fair because the nominal figure moves upward, even though purchasing power drops.

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Fairness in Product Pricing: Caselets

First National Bank of Chicago (mid-1990s)

To cut costs, FNB Chicago imposed a $3 fee per teller transaction (intended to push customers to ATMs). Public outrage: front-page headlines (“First Chicago loses touch with humans”), competitors advertised “free teller” service, radio ads mocked the fee. In 2002 the fee was removed.

Alternative strategy: Instead of penalizing teller use, the bank could have rewarded ATM use (e.g., $1 bonus per ATM transaction). Fairness perception differs between a penalty and a foregone reward — even if the net effect is identical.

Coca-Cola’s dynamic pricing proposal (late 1990s)

CEO Douglas Ivester argued that vending machines should charge higher prices on hot days, when willingness to pay spikes. The press attacked this as price gouging. New York Times and Wall Street Journal criticised the plan; Ivester later resigned, partly due to the backlash.

Key insight: Standard economics says price should rise with demand. But consumers see it as unfair when firms exploit transient needs — even when marginal cost is constant.

Whitney Houston album price hike (2012)

Hours after Whitney Houston’s death, iTunes and Sony raised prices of her albums on UK iTunes:

• The Ultimate Collection: £4.99 → £7.99 (+60%)
• Whitney: The Greatest Hits: £7.99 → £9.99 (+25%)

Consumers were furious: “I am angry is an understatement … totally parasitic.” Digital copies have zero marginal cost; there was no shortage. The price increase solely exploited a demand spike, violating fairness norms.

Why are some industries treated differently?

The first mover often takes a hit. Once a new price norm becomes socially accepted — as in the airline industry — consumers stop perceiving it as unfair.

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Surge Pricing and the Uber Controversy

Uber’s surge pricing multiplies fares (2×, 5×, 10×) when demand exceeds supply. The company’s defence: higher prices attract more drivers, rebalancing the market.

Criticisms:

• The formula is proprietary (“invisible hand visible only to Uber”)
• Data on driver labour supply elasticity is undisclosed
• Drivers cannot instantly respond to surge pricing; peak hours are routine, so drivers are already on the road
• Sustained high surges (5×–10×) suggest the mechanism fails to bring on additional drivers

During Hurricane Sandy, New York Attorney General Eric Schneiderman accused Uber of price gouging. Uber later took corrective steps to address state concerns.

Practical exercise: Ask ride-hailing drivers how they feel about surge pricing on weekend evenings vs. during a cyclone — fairness judgments differ by context.

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Key Takeaways

• Downward nominal wage rigidity explains why recessions cause job losses rather than wage cuts; workers’ fairness perceptions prevent nominal reductions.
• Fairness is evaluated in nominal terms — a raise in nominal wages (even if real wages fall) is acceptable; a cut is not.
• Fairness violations in pricing can destroy brand reputation and force CEOs to resign (First Chicago, Coca-Cola, Whitney Houston case).
• Firms that move first to impose “unfair” pricing face backlash; once a new social norm sets in (e.g., airline fees), the practice becomes accepted.
• Surge pricing remains controversial because of opacity and limited effectiveness in adding supply during predictable peak times.