---
title: "Choosing"
book: "The Industry: Firms, Roles and Careers"
subject: quant
language: en
chapter: 30
exercises: 8
source: https://one-course.com/books/quant/17/en/chapter/30-choosing
---

# Chapter 30 — Choosing

Four offers, a bank quant role, a market maker’s trading role, an analyst seat at a platform and a technology firm’s engineering job, differ in expected pay, in its spread, in what tax and housing take, in hours, in what the next five years teach and in how likely the job is to last. No single number ranks them: a person’s weights do, and weights are hard to state. The useful questions are which offers no weights can save, how often each offer comes first when the weights are uncertain, and how far a weight must move before the choice changes. This chapter builds that analysis from the tools of the previous chapters and closes the book.

## 30.1 The attributes that differ across firm types

The chapters of Parts II and III measured what differs.

- **Pay** , its structure and its spread (chapters 13 and 14): base and bonus, deferral and [forfeiture](https://one-course.com/books/quant/17/en/chapter/13-how-pay-works#def-in-how-pay-works-rsu) , formulaic payouts, and the ranges the filings show.
- **Place** (chapters 15 and 27): tax, housing and price levels.
- **Hours** (chapter 26): the coverage a desk needs and the on-call it shares.
- **The work** (chapters 16 to 25): what a role ships, how fast it learns whether it is good (chapter 17’s [feedback speed](https://one-course.com/books/quant/17/en/chapter/17-quantitative-researcher#def-in-quant-researcher-feedback) ) and what it leads to (chapter 28).
- **Security** : the stops of a platform (chapter 22) and the chance that a job ends for reasons the holder does not control.

**Definition 30.1 (Multi-attribute value model, swing weighting).**

A *multi-attribute value model* scores each alternative on each attribute, scales every attribute from the worst to the best alternative, and adds the scaled scores with weights (Keeney and Raiffa). *Swing weighting* sets the weights by asking how much the chooser values moving each attribute from its worst to its best level, relative to the others, rather than how important the attribute is in the abstract.

## 30.2 Pay risk as a certainty equivalent

Chapter 13 turned a package’s distribution into a certainty equivalent: the sure amount with the same expected utility under constant relative risk aversion (Book 4, chapter 9). With relative risk aversion 3 and $500 000 of other wealth, the chapter’s four offers are worth $1 186 354 (bank quant), $1 365 186 (market maker), $903 603 (platform analyst) and $845 964 (technology engineer) over five years. The certainty equivalent folds risk into pay; the value model below keeps them apart, as expected pay and pay risk (the 10th–90th percentile range over the mean), so that the weight on risk can be seen and varied.

| offer (city) | expected pay | pay risk | tax, housing | on-call | learning | security |
| --- | --- | --- | --- | --- | --- | --- |
|  | $k, 5 years | range/mean | share | nights/month | 1–5 | share |
| bank quant (London) | 1 618 | 1.01 | 53.1% | 0.0 | 3 | 1.00 |
| market maker (Chicago) | 1 949 | 1.16 | 40.8% | 3.8 | 4 | 1.00 |
| platform analyst (New York) | 1 703 | 1.67 | 50.0% | 0.0 | 4 | 0.65 |
| tech engineer (London) | 1 191 | 1.22 | 55.3% | 5.1 | 3 | 1.00 |

The table’s pay figures come from chapter 13’s offers and a fourth built the same way (a technology firm’s base with a restricted-stock grant vesting over four years); every package is illustrative. The share taken by tax and housing applies chapter 27’s tools to each offer’s average annual pay in its city; on-call nights use chapter 26’s rotas (eight people for the market maker’s round-the-clock desk, six for the engineer’s service); learning is the chooser’s judgement; security is the share of five-year careers in which the pay model does not close the job, which only the platform’s book can do.

## 30.3 Temperament, skills and the work itself

Some attributes resist numbers and matter most. Chapter 17’s [feedback speed](https://one-course.com/books/quant/17/en/chapter/17-quantitative-researcher#def-in-quant-researcher-feedback) decides how soon a person learns whether they are good at the job; chapter 16’s supervision load and chapter 26’s hours decide what a day is like; chapter 22’s stops decide how much a bad year costs. A person who dislikes uncertain pay should say so in the weight on pay risk; a person who wants to build systems rather than trade them should say so in the learning score. The model does not know temperament; it makes the chooser state it and shows what follows.

## 30.4 A multi-attribute framework and its sensitivity

**Definition 30.2 (Dominated alternative, rank acceptability).**

An alternative is a *dominated alternative* when another is at least as good on every attribute and better on one: no weights can make it the best. The *rank acceptability* of an alternative for a rank is the share of weight vectors, drawn from a stated distribution, under which the alternative takes that rank (Lahdelma, Hokkanen and Salminen’s stochastic multicriteria acceptability analysis).

**Method 30.3 (Screening, weighting and sensitivity).**

Scale each attribute to $[0,1]$ from the worst to the best offer (reversed where less is better). Drop dominated offers. With swing weights $w$, the value of offer $i$ is $\sum_j w_j s_{ij}$. Draw weights uniformly on the simplex (a Dirichlet distribution with unit parameters) and count how often each offer takes each rank. For the attribute of interest, find the smallest change of its weight, the others rescaled in proportion, that changes the best offer.

With the illustrative swing points (expected pay 30, pay risk 15, place 10, on-call 15, learning 20, security 10), the market maker’s offer scores 0.852, the platform’s 0.589, the bank’s 0.584 and the technology engineer’s 0.202. The technology offer is dominated: the bank’s offer pays more with less spread, takes less in tax and housing, asks for no on-call, and matches it on learning and security. Under uniformly random weights ([Figure 30.1](#fig-in-choosing-ranks)) the market maker comes first in 79.5% of draws, the bank in 15.9%, the platform in 4.6% and the technology offer never.

![Rank acceptability of the four illustrative offers: the share of 20 000 weight vectors, drawn uniformly on the simplex of six attributes, under which each offer takes each rank. Data: in_choose.results, from firm.careerdec.rank_acceptability.](https://one-course.com/images/onecourse/chapters/quant-17/in-choosing/fig-c36f99db5ea5.svg)

***Figure 30.1.** [Rank acceptability](#def-in-choosing-smaa) of the four illustrative offers: the share of 20 000 weight vectors, drawn uniformly on the simplex of six attributes, under which each offer takes each rank. Data: `in_choose.results`, from `firm.careerdec.rank_acceptability`.*

The market maker’s lead is robust to the weight on pay risk: that weight must rise by 0.452, from 0.15 to 0.602 with the other weights shrunk in proportion, before the bank’s offer becomes the best ([Figure 30.2](#fig-in-choosing-sweep)). A chooser for whom the spread of pay outweighs expected pay, place, hours, learning and security together should take the bank’s offer; anyone else, on these numbers, the market maker’s.

![Each offer’s value as the weight on pay risk varies, the other weights rescaled in proportion. The dashed lines mark the chosen weight (0.15) and the weight at which the bank’s offer overtakes the market maker’s (0.602). Data: in_choose.results and firm.careerdec.flip_shift.](https://one-course.com/images/onecourse/chapters/quant-17/in-choosing/fig-5d6235dbdad8.svg)

***Figure 30.2.** Each offer’s value as the weight on pay risk varies, the other weights rescaled in proportion. The dashed lines mark the chosen weight (0.15) and the weight at which the bank’s offer overtakes the market maker’s (0.602). Data: `in_choose.results` and `firm.careerdec.flip_shift`.*

## 30.5 Revisiting the decision

A decision about a job is revisited, and the revisiting is where most mistakes happen. Book 16, chapter 26, describes the tools: a decision journal written at the time, which records the attributes, weights and expectations, and so guards against outcome bias (judging the choice by how the year went rather than by what was known); and a pre-mortem, which asks before the move how it could fail. Chapter 28’s move model says when leaving is expensive, and chapter 17’s [feedback speed](https://one-course.com/books/quant/17/en/chapter/17-quantitative-researcher#def-in-quant-researcher-feedback) says how long to wait before judging a research job. The model of this chapter is a record of what the chooser believed; its value is in being rerun when a belief changes.

## 30.6 Tutorial: four offers

**Goal.** Score four offers from the earlier chapters’ tools, screen, weight, and measure how robust the choice is. **End state:** Figures [30.1](#fig-in-choosing-ranks) and [30.2](#fig-in-choosing-sweep) and the attribute table.

1. **Attributes.** `firm.payoffer` for pay and its spread; `firm.locations` for what tax and housing take; `firm.workload.oncall_nights` ; judgements for learning; the pay model’s closures for security.
2. **Scale and screen.** `firm.careerdec.scale` and `dominated` .
3. **[Rank acceptability](#def-in-choosing-smaa).** `rank_acceptability` draws weights on the simplex ([Listing 30.1](#lst-in-choosing-smaa)). `def rank_acceptability (scaled, n, rng): s = np.asarray(scaled, float ) w = rng.dirichlet(np.ones(s.shape[1 ]), n) v = w @ s.T # (n, offers) ranks = (-v).argsort(axis=1 ).argsort(axis=1 ) # 0 = best k = s.shape[0 ] return np.stack([(ranks == r).mean(axis=0 ) for r in range (k)], axis=1 )` **Listing 30.1.** Rank acceptability by sampling weights uniformly on the simplex. code/firm/careerdec/firm_careerdec.py
4. **Sensitivity.** `flip_shift` searches the smallest change of one weight that changes the best offer ([Listing 30.2](#lst-in-choosing-flip)). `def flip_shift (scaled, weights, attr, step=0.001 ): """Smallest |change| of weights[attr] (the others rescaled in proportion) that changes the best offer; returns (signed change, new best index) or None if no change in [0, 1] does.""" best = int (np.argmax(value(scaled, weights))) w0 = float (np.asarray(weights)[attr]) for k in range (1 , int (1.0 / step) + 1 ): for sign in (1 , -1 ): new = w0 + sign * k * step if 0.0 <= new <= 1.0 : b = int (np.argmax(value(scaled, _shifted(weights, attr, new)))) if b != best: return sign * k * step, b return None` **Listing 30.2.** The smallest change of one weight that changes the choice. code/firm/careerdec/firm_careerdec.py

The results: first-rank acceptability 79.5% (market maker), 15.9% (bank quant), 4.6% (platform analyst), 0% (technology engineer); the weight on pay risk must rise by 0.452 to change the choice.

**What to change next.** Replace the uniform weight distribution with the chooser’s ranking of attributes (ordinal SMAA); add correlation between attributes’ uncertainty; run the analysis again after a year with what was learnt.

## 30.7 Build: firm.careerdec

**Purpose.** Choose among offers with several attributes, and say how robust the choice is to the weights.

**Interface.** `firm.careerdec`: `scale(matrix, higher_better)`; `swing_weights(points)`; `value(scaled, weights)`; `dominated(matrix, higher_better)`; `rank_acceptability(scaled, n, rng)`; `flip_shift(scaled, weights, attr, step)`; fed by `firm.payoffer`, `firm.aftertax`, `firm.locations`, `firm.workload` and `firm.careerpath`.

**Rules.** Scales run from the worst to the best offer; dominance is checked on raw values; weights sum to one; the flip search changes one weight and rescales the others in proportion.

**Acceptance tests.** `code/firm/careerdec/tests/`: scaling and reversal; a dominated offer is found; two mirror-image offers each come first half the time; a flip is found where it must be and not where it cannot be.

**Stretch.** Ordinal weights; interval scores; a value function with interactions between attributes.

Sources and further reading

- Keeney, R. L. and H. Raiffa (1976), *Decisions with Multiple Objectives: Preferences and Value Tradeoffs* , Wiley (Cambridge University Press, 1993).
- von Winterfeldt, D. and W. Edwards (1986), *Decision Analysis and Behavioral Research* , Cambridge University Press.
- Lahdelma, R., J. Hokkanen and P. Salminen (1998), SMAA—stochastic multiobjective acceptability analysis, *European Journal of Operational Research* 106(1), 137–143.
- Chapters 13, 15, 17, 22 and 26–28 of this book; Book 4, chapter 9; Book 16, chapter 26.

## 30.8 Exercises

**Exercise 30.1 ★.**

Which offer is dominated, and by which?

**Solution of Exercise 30.1.**

The technology engineer’s offer, by the bank quant’s: more expected pay, less spread, a smaller share taken by tax and housing, no on-call, and equal learning and security.

**Exercise 30.2 ★.**

What are the swing weights, as shares, of the six attributes?

**Solution of Exercise 30.2.**

Expected pay 0.30, pay risk 0.15, place 0.10, on-call 0.15, learning 0.20, security 0.10.

**Exercise 30.3 ★.**

Which offer has the highest certainty equivalent at relative risk aversion 3, and which the highest expected pay?

**Solution of Exercise 30.3.**

The market maker’s offer on both: a certainty equivalent of $1 365 186 and expected pay of $1 949 485 over five years.

**Exercise 30.4 ★★.**

Why does the platform offer come first in only 4.6% of weight draws although it ties for the best learning score?

**Solution of Exercise 30.4.**

It has the widest spread of pay and the only chance of a closed job, the worst scores on two attributes, and it trails the market maker on expected pay and on place; only draws that weight learning and no-on-call heavily and pay risk and security lightly put it first.

**Exercise 30.5 ★★.**

What is the weakness of drawing weights uniformly on the simplex?

**Solution of Exercise 30.5.**

It treats every weighting as equally likely, including extreme ones no real chooser holds, and ignores what the chooser knows about her own priorities; the acceptabilities describe the offers’ robustness, not her preference.

**Exercise 30.6 ★★.**

A friend weights learning at 60 swing points and the rest as in the chapter. Which offer does the model choose?

**Solution of Exercise 30.6.**

Still the market maker’s offer (0.894, against 0.706 for the platform’s): it ties for the best learning score.

**Exercise 30.7 ★★★.**

*Coding.* Drop the dominated offer and recompute the first-rank acceptability of the other three.

**Solution of Exercise 30.7.**

Rescaled over three offers, the first-rank acceptabilities become 81.4% (market maker), 13.5% (bank quant) and 5.1% (platform analyst): dropping a dominated offer changes the scales and so, a little, the shares.

**Exercise 30.8 ★★★.**

*Find the flaw.* “The market maker’s offer wins in 79.5% of weightings, so it is the right choice for 79.5% of people.”

**Solution of Exercise 30.8.**

The weights are drawn uniformly by assumption, not from a survey of people; the index measures how much of the weight space favours an offer, which says nothing about how many people hold which weights. And the attribute values are illustrative.

## 30.9 Problem: Four Offers

**Problem 30.1.**

Weekend problem — four offers

A graduate holds the four offers of the chapter and asks which to accept.

**Part I — The attributes.**

1. Which attributes differ across the offers, and which chapters measure each?
2. Define the [multi-attribute value model](#def-in-choosing-mavm) and [swing weighting](#def-in-choosing-mavm) .
3. State the four certainty equivalents at relative risk aversion 3.
4. Why keep expected pay and pay risk apart?
5. Which attributes are judgements?

**Part II — The model.**

6. Give the attribute table.
7. Define a [dominated alternative](#def-in-choosing-smaa) and name the dominated offer.
8. State the swing points and the offers’ values.
9. Define [rank acceptability](#def-in-choosing-smaa) and state the method.
10. Give the first-rank acceptabilities.

**Part III — Sensitivity.**

11. How far must the weight on pay risk move to change the choice?
12. What does that say about the bank’s offer?
13. What would change the learning scores?
14. What would a larger platform payout change?
15. What does the model not know?

**Part IV — The verdict.**

16. State the *named result* : each offer’s first-rank acceptability index under uniform weights, and the smallest change in the weight on pay risk that changes the preferred offer.
17. How should the graduate record the decision?
18. When should she revisit it?
19. What would make her regret it, and how would she tell bad luck from a bad choice?
20. In two sentences, answer her.

**Solution of Problem 30.1.**

1. Pay and its spread (13, 14), place (15, 27), hours (26), the work and what it leads to (16–25, 28), security (22).
2. As in the chapter’s definition.
3. $1 186 354, $1 365 186, $903 603 and $845 964.
4. So that the weight on risk is explicit and can be varied.
5. Learning, and the weights themselves.
6. As in the chapter’s table.
7. As in the definition; the technology engineer’s offer.
8. 30, 15, 10, 15, 20, 10; values 0.852 (market maker), 0.589 (platform), 0.584 (bank), 0.202 (technology).
9. As in the definition; scale, screen, weight, sample weights on the simplex and count ranks.
10. 79.5%, 15.9%, 4.6% and 0%.
11. By 0.452, from 0.15 to 0.602.
12. It is the choice only for someone who weights pay risk above all the other attributes together.
13. Evidence about what each job teaches: its [feedback speed](https://one-course.com/books/quant/17/en/chapter/17-quantitative-researcher#def-in-quant-researcher-feedback) , its training, where its people go next.
14. It would raise the platform’s expected pay and spread together.
15. Temperament, the people, and the chooser’s circumstances.
16. Market maker 79.5%, bank 15.9%, platform 4.6%, technology 0%; the weight on pay risk must rise by 0.452.
17. In a decision journal: attributes, weights, expectations and what would change her mind.
18. When a belief changes (pay, learning, security), and at a date fixed in advance, not after a bad month.
19. A result she could not have foreseen is bad luck; a result she could have foreseen and ignored is a bad choice; the journal tells them apart.
20. On the chapter’s numbers the market maker’s offer wins under most weightings, and only a very strong aversion to pay risk favours the bank’s. Record why, and revisit when a belief, not an outcome, changes.

## 30.10 Interview questions

**Interview question 30.1 ★ researcher, trader.**

Why do you want this job rather than the others you are considering?

**Solution of Interview question 30.1.**

Name the attributes that decide it for you (the work, what you will learn, the people) and show you know the job’s realities; do not run down the other offers.

*What the interviewer is looking for: specific reasons tied to the job.*

**Interview question 30.2 ★ researcher.**

How would you compare two offers whose pay differs in structure but not in expected value?

**Solution of Interview question 30.2.**

By risk and timing: certainty equivalents under a stated risk aversion, deferral and [forfeiture](https://one-course.com/books/quant/17/en/chapter/13-how-pay-works#def-in-how-pay-works-rsu) (what leaving costs), and tax timing.

*What the interviewer is looking for: structure matters even at equal expected value.*

**Interview question 30.3 ★★ researcher, developer.**

Write a function that finds which alternatives are dominated. What is its complexity?

**Solution of Interview question 30.3.**

Compare each pair: offer $i$ is dominated if some $j$ is at least as good everywhere and better somewhere; $O(n^2m)$ for $n$ offers and $m$ attributes, fine for small $n$; a sort-based skyline algorithm does better for large $n$.

*What the interviewer is looking for: the pairwise definition and its cost.*

**Interview question 30.4 ★★ researcher.**

How would you sample weights uniformly on a simplex?

**Solution of Interview question 30.4.**

Draw independent exponential variables and normalise them (a Dirichlet with unit parameters), or sort uniform draws and take the gaps; normalising uniform draws is not uniform on the simplex.

*What the interviewer is looking for: the Dirichlet construction and the common mistake.*

**Interview question 30.5 ★★ trader.**

You took a job that went badly in its first year. How do you decide whether it was a bad decision?

**Solution of Interview question 30.5.**

Compare the outcome with what was known and expected when choosing (the journal): if the bad result was a foreseeable risk you accepted knowingly, it was bad luck; if you ignored information you had, it was a bad decision.

*What the interviewer is looking for: avoiding outcome bias.*

**Interview question 30.6 ★★★ researcher.**

Two attributes are strongly correlated across your options. What does that do to an additive value model, and what would you do about it?

**Solution of Interview question 30.6.**

The additive model double counts what the two attributes share, giving it too much weight; merge them, reduce the weights of both, or keep only the one that matters to you.

*What the interviewer is looking for: double counting in additive models.*