Exercise Bank — Emergence and Artificial Societies

The printed Chapter 15 keeps Exercises 1–5; the bank continues from Exercise 6.

Exercise 6. Play Section 15.3’s naming game by hand before playing it at scale. (a) Three agents 0, 1, 2 start with empty inventories and meet in the scripted order (0 \to 1), (2 \to 0), (0 \to 2), (1 \to 2), (0 \to 1), speaker first; a speaker holding several words utters the alphabetically first (the module’s draw, pinned); invented words are w_0, w_1, \ldots in order of coinage; success and failure follow foundations/emergence/naming_game.py. Trace every inventory round by round, give the round and word of convergence, and record the population’s total vocabulary \sum_i |I_i| after each round, where I_i is agent i’s inventory — babel grows before it collapses; explain also why it is safe for the module to test for consensus only after a success. (b) With the module, measure the mean rounds to convergence over seeds 0–29 for populations n \in \{10, 20, 40, 80\}, and estimate the exponent \beta in \mathrm{rounds} \sim n^{\beta}; then price the finding: extrapolating at your fitted \beta, roughly how many pairwise exchanges would a hundred-agent fleet spend self-organising one convention that Section 15.3’s first route — design it offline and distribute it — delivers for a single broadcast? (c) Extend the module: m of n = 50 agents are committed — they hold “z”, always utter “z”, and never update — while the remaining 50 - m start from an established convention “w”; for m \in \{1, 2, 3, 4, 5, 6, 8, 10\} and twenty seeds each, measure how often the non-committed population has fully tipped to “z” within 30{,}000 rounds, and how fast when it does; locate the tipping window, explain why a run that fails to tip within the cap evidences metastability rather than impossibility, and connect the curve to the committed-minority finding of Section 15.4 — Schelling’s tipping, operating in belief space.

Exercise 7. Section 15.3‘s honesty clause, made runnable. Two micro-stories, four hundred agents each, every agent starting at wealth 1.0, stdlib and seeded throughout. Exchange: for 200{,}000 encounters, draw a random pair, pool the two agents’ wealth, and split the pool at a uniformly random point. Shocks: for 90 rounds, multiply every agent’s wealth independently by 1.1 or 0.9 on a fair coin — no agent ever interacts with another. (a) Implement both and report, for each, the Gini coefficient, the mean-to-median ratio, and the top decile’s share of total wealth. (b) Both stories hit the same macro-target — a right-skewed distribution with a long rich tail and Gini near 0.48 — and Section 15.3‘s Sugarscape grows the same shape from a third mechanism, positional advantage compounding; state exactly what the first successful growth proved, in the credo’s own terms, and what it did not. (c) Compute a held-out statistic neither story was tuned to, the pauper rate \Pr(w < \bar{w}/20): report it for both stories, check the two values against the closed forms for the stories’ limiting distributions (exponential and lognormal), and say which end of the distribution the rival micro-stories actually disagree about. (d) In three sentences: name the discipline that (c) instantiates, and propose one real-world observable that would decide between exchange and shocks for an actual economy.

Exercise 8. The deployed fleet as a society, priced. A review fleet contains twelve agents; a prompt regression strikes a model lineage in any given week with probability q = 0.1, independently across lineages, and when it strikes it shifts the behaviour of every copy of that lineage at once. (a) Compare a monoculture (twelve copies of one lineage) with a mixed fleet (six copies each of two lineages): compute, for both, the expected number of affected agents in a week, the probability that the whole fleet shifts simultaneously, and the probability that at least half of it does; then state the exact sense in which diversity helps, and the sense in which it does not. (b) Five reviewers vote on each verdict, each correct with probability 0.7: compute the majority’s accuracy when the five are independent and when they are five copies of one model; then, under a common-draw model in which the panel collapses onto one shared draw with probability \rho and votes independently otherwise, find the \rho at which the five-strong panel is worth exactly three independent reviewers. (c) Normal per-agent spend on a task is 800 tokens with standard deviation 300, independent across agents, and each agent’s individual circuit breaker trips at 2{,}000 tokens; a herding episode has all twelve agents burning 1{,}800 each. Show that no individual breaker trips; compute how many standard deviations the episode’s aggregate spend sits above the aggregate’s mean; give the trip point of an aggregate breaker set at mean plus four standard deviations, and its false-alarm probability under a normal approximation — the stampede in which every animal is individually within limits, quantified. (d) Map your three results onto Table 15.3, one row each.

Exercise 9 (lab). Replicate the naming game’s arrival in silicon, in miniature. Fix a pool of ten short names and a population of six agents, each an instance of one live model; in each round draw a distinct pair, give each of the two only the name pool (order freshly shuffled per call, to guard against position bias) and its own last five interactions — what it played, what its partner played, whether they matched — and ask it to play the one name it expects its partner to play; the round succeeds when the two names match, and both agents record the outcome. Run to twelve consecutive successes all on the same name — assorted matches are luck, not a convention — or eighty rounds, whichever comes first; at stop, take a census (one fresh call per agent: which name would you play?) and claim convergence only when the census is unanimous, a pair-streak sampling pairs rather than the population; complete at least five runs. Separately measure the individual prior: twenty fresh single-shot calls, each asked once which name from the (again shuffled) pool it would choose. Report each run’s convergence round and winning name; set the distribution of winners across runs against the single-shot distribution and say whether the population exhibits a collective bias in Section 15.4’s sense — a preference of the population that no member exhibits alone, or a sharpening of one that some do; then test one of the chapter’s cautions by rerunning one condition on a different model and noting what survives the swap. Record the model identifiers and the date beside the tables: whether conventions crystallise at all, and the gap between winners and priors, are the durable findings — the specific winning name is weather.

Exercise 10 (lab). Build Exercise 4’s cascade machine out of live reviewers and watch the arithmetic come true, or fail to. One patch, actually broken; eight reviewer instances; each holds one private evidence card — a short test transcript, either misleadingly green (the suite passed on a stale build) or honestly red (the failure reproduced) — scripted so that reviewers 1 and 2 hold green cards and reviewers 3–8 hold red ones. Three regimes, at least ten trials each, fresh instances per trial: (i) sequential — each reviewer sees only its own card plus its predecessors’ bare verdicts, and answers “approve” or “reject”; (ii) independent — each sees only its own card, and the eight verdicts are majority-voted; (iii) contrarian — as (i), except reviewer 5’s card is upgraded to a complete failing run with the root cause named, and its verdict is permitted one sentence of evidence. Classify every verdict as follows-card (it matches the agent’s own evidence) or follows-crowd (it contradicts that evidence in the direction of the predecessors’ majority), tabulate by position and regime, and answer three questions: does regime (i) reproduce Exercise 4’s rational herding — approvals from reviewer 3 onward — or does the model weight its own card more or less than Bayes licenses; by how much does regime (ii) beat regime (i) on the final decision about the patch; and in regime (iii), do reviewers 6–8 abandon the crowd once a single visibly better-informed dissent is on the record, as the fragility clause predicts? Record the model identifier and the date beside the table: the durable findings are the orderings — independent versus sequential, shatter versus hold — not any absolute rate.