23506 Chap.12 The Maths Toolkit: Interval vs Set Notation, Expected Value and Discounted Infinite Sums
The Maths Toolkit: Interval vs Set Notation, Expected Value and Discounted Infinite Sums
Mathematics in 23506 records feasible choices, information and incentives. Intervals such as [0,100] include both boundaries, while (0,1) excludes them. Parameter solutions should partition the entire feasible domain without gaps or overlap. Equality points are strategic because an argmax can contain several actions there, increasing the equilibrium set. Exact fractions preserve those ties better than premature decimals.
Expected payoff weights mutually exclusive unresolved states by probabilities summing to one. Insurance without cover yields 10α+100(1−α)=100−90α and is compared with guaranteed 80, producing α≥2/9. Oil investment 5α+(1−α)(−2)=7α−2 preserves the failure loss and produces 2/7. After a state is observed, use the realised payoff instead of averaging again. In mixed games, probability ownership crosses players because one player’s randomisation creates the other’s uncertainty.
Functions encode contingent strategies such as follower responses after every leader quantity. Argmax returns the action set attaining the maximum, while maximum is the payoff value. Discount factors convert dated payoffs: z received k delays later is δ^kz; k forever from now is k/(1−δ), and from next period δk/(1−δ). Inequality direction follows sign: multiplying by positive 1−δ is safe, while dividing by a negative coefficient reverses direction. Every solution should be checked against its domain, endpoints, units and the original equality.
What this chapter covers
- 01Intervals, sets and equality points
- 02Functions and contingent strategies
- 03Expected value before unresolved states
- 04Mixed expected payoff and probability ownership
- 05Inequalities, argmax and piecewise responses
- 06Finite and infinite discounted sums
Insurance response correspondence
- 2Define the probability domain and complement.
- 2Write and simplify uninsured expected payoff.
- 2Compare insurance and uninsured values with the correct inequality.
- 2Solve the exact cutoff and verify equality by substitution.
- 2State unique actions on each open region and both actions at equality.
Key terms
- Closed interval
- An interval including an endpoint marked with a square bracket.
- Expected value
- A probability-weighted average over mutually exclusive unresolved outcomes.
- Probability complement
- For a binary event of probability α, the other event has probability 1−α.
- Argmax
- The complete set of feasible actions attaining the highest value.
- Present value
- A dated payoff translated to a chosen evaluation date through discounting.
- Boundary case
- A parameter equality where preference can become weak or set-valued.
The Maths Toolkit: Interval vs Set Notation, Expected Value and Discounted Infinite Sums FAQ
What does [a,b] mean?
Both a and b are included; parentheses exclude the corresponding endpoint.
How should oil probability be partitioned?
Use strict regions separated by singleton cutoffs 1/5 and 2/7, then include domain endpoints 0 and 1 appropriately.
Why retain a minus sign in expected payoff?
A negative terminal payoff is a real loss; deleting it changes the intercept, slope and strategic threshold.
When is expectation appropriate?
When the decision is made before the relevant state or opponent action is observed.
Who is made indifferent in a mixed equation?
The player facing the probability; that opponent’s indifference solves the probability owner’s mix.
What is the difference between maximum and argmax?
Maximum is the best value; argmax is the action or actions producing it.
When does an inequality reverse?
When both sides are multiplied or divided by a negative number.
Why is δ² used for a period-3 payoff from period 1?
Two one-period delays separate the evaluation and delivery dates.
What is a stream beginning next period worth?
δk/(1−δ), with the leading δ marking the first delay.
What checks should follow a calculation?
Domain, probability sum, units, sign, exact equality and strategic interpretation on both sides.
What is the full mathematical error-check routine?
Define each symbol, its owner, feasible domain, payoff unit, information date and delivery date. For probabilities, use complements that sum to one and evaluate endpoints to recover pure-state payoffs. Parenthesise negative outcomes before expanding so a failure loss cannot lose its sign. For a mixed strategy, label whose probability is being solved and whose payoffs are equated; confirm all probabilities are non-negative, sum to one and make every supported action attain the same maximum. For inequalities, preserve the original comparison, record the sign of every factor used to multiply or divide, and substitute the exact cutoff back into both payoffs. Partition the feasible domain with no gap or overlap and state all actions at equality. For discounting, write the first three dated terms so the leading exponent is visible; distinguish a finite sum, a stream beginning now and one beginning next period. Finish by translating argmax into an action sentence and maximum into a payoff sentence. Exact fractions remain primary; decimals are interpretation aids only.
Exam move
Maintain a symbol dictionary containing variable meaning, owner, domain, payoff unit and information date. For intervals, sort exact thresholds and test domain endpoints. For expectation, verify weights sum to one and substitute pure-probability endpoints. For mixed strategies, label whose expected payoff appears in each equation. For inequalities, retain the original comparison, note the sign of every divisor and substitute the cutoff into unsimplified payoffs. For discounting, write the first three terms before using a closed form so the leading exponent is visible. Convert every final symbol into a sentence naming the optimal action below, at and above a boundary. Use a calculator only after the symbolic setup, keeping exact fractions in the answer.
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