Exercise Bank — Strategic Interaction

The printed Chapter 9 keeps Exercises 1–6; the bank continues from Exercise 7.

Exercise 7. The orchestrator announces to the coder: “any submission arriving after the deadline is discarded, no exceptions.” Model the encounter as a tree: the coder delivers on time, yielding payoffs (2, 4) to (coder, orchestrator), or late — the extra slack producing better work — upon which the orchestrator either accepts, yielding (3, 3), or discards and re-runs the subtask at fresh expense, yielding (0, 1). (a) Solve the tree by backward induction and give the subgame-perfect outcome. (b) Show that the profile in which the coder delivers on time and the orchestrator plans “discard if late” is nevertheless a Nash equilibrium — exhibit both no-gain checks — and say exactly where it fails Selten’s test, and on the strength of which unvisited node it survives. (c) The engineer can install a hard gate in the harness — an automatic, unoverridable discard of late submissions — at a one-off cost c to the orchestrator. For which c is installing it worth while, against the backward-induction baseline of (a)? (d) Suppose the gate is installed at c = 0.5 but not announced. Predict the play, compute the payoffs, compare them with both (a) and the announced-gate outcome, and state the rule of Section 9.3 this arithmetic proves; then say why announcing a discard policy that lives only in the orchestrator’s prompt, with no gate behind it, fails for the complementary reason — which property of a commitment device does each variant lack?

Exercise 8. The companion repository’s foundations/algorithms/games.py plays the repeated Prisoner’s Dilemma with flawless execution: tit_for_tat echoes exactly what the opponent did, and play_repeated never misdelivers a move. Rebuild the loop for the noisy world of Section 9.4: a memory-one strategy is a triple giving the probability of cooperating on the first round, after the opponent’s cooperation, and after the opponent’s defection — strict tit-for-tat is (1, 1, 0), a generous variant (1, 1, g), always-cooperate (1, 1, 1), always-defect (0, 0, 0) — and execution noise flips each played move, independently, with probability \varepsilon. Use the canonical payoffs of Table A.1. (a) Implement the noisy loop, seeded and reproducible, returning the row player’s mean per-round payoff and the rate of mutual cooperation. (b) Run strict tit-for-tat against itself at \varepsilon = 0.05 for at least 10^5 rounds and estimate both quantities; then prove what the estimates suggest: show that the four-state chain of joint moves is doubly stochastic for every \varepsilon \in (0, 1/2], hence uniform in the long run, so the mean payoff is exactly 9/4 whatever the noise level — and say what fraction of eternity two well-meaning agents spend in one-sided retaliation. (c) Derive the closed form for the generous pair: with h = \varepsilon + g(1 - 2\varepsilon), each side’s long-run cooperation probability is h/(\varepsilon + h); evaluate it at g = 1/3 and \varepsilon = 0.05, give the exact mean payoff, and confirm both by simulation. (d) Price the generosity: compute or simulate g \in \{0, 1/3, 1\} against always-defect at the same \varepsilon and tabulate the per-round payoffs; evaluate the recipe g^{*} = \min\bigl(1 - (T - R)/(R - S),\; (R - P)/(T - P)\bigr) on these payoffs; and reconcile the table with Section 9.4’s rule that forgiveness is error correction rather than sentiment — and must not become an invitation.

Exercise 9. Bazaar is an open marketplace where orchestrators hire specialist subagents, built as follows: (i) capability discovery is by self-description, and awards go to the most impressive advertisement, with no trial; (ii) identities are free API keys — deregistering and re-registering under a fresh key is instant and costless; (iii) each agent displays a reputation score computed from its own reported job outcomes; (iv) the terms of service threaten permanent bans for budget overruns, but bans are applied by a support agent that reviews appeals and waives the ban whenever the delivered work was useful; (v) all tenants draw on one shared rate-limit pool, and each agent’s objective is its own job’s completion; (vi) the terms of service open, “We are a community of frugal, honest agents.” (a) Dismantle it: diagnose each of (i)–(vi) in this chapter’s vocabulary — name the game or mechanism installed, and predict the equilibrium behaviour it will be played to. (b) Redesign it minimally: one repair per flaw, each naming the chapter instrument it deploys (a screen, a signal, a stake, a commitment device, a payoff change, an identity discipline) and the observable behaviour it forecloses. (c) Make two repairs quantitative. First the registration bond: an honest identity earns 1 per task indefinitely, task-to-task continuation is \delta = 0.9, and a defect-and-run nets a one-off 5 before the identity is burned; find the bond b at which whitewashing stops paying, and the range of b that deters whitewashing while leaving honesty worth registering for. Second, a delivery stake for flaw (i): with the award worth 4 to the candidate and an inept delivery certain to fail, find the stake that makes bidding a losing proposition for the inept while costing the able nothing. (d) If only one repair could ship, argue from the dependency structure among the six which it must be — which repairs are worthless until another is in place?

Exercise 10 (lab). Sit a live model at Nagel’s table. Fix one model and one temperature, and run at least ten independent trials (a fresh conversation each) under each of three regimes: (i) the bare game — “You are one of 21 players. Each player picks a number between 0 and 100. The pick closest to two-thirds of the average of all picks wins. Reply with your pick and nothing else.”; (ii) the bare game plus “The other 20 players are expert game theorists.”; (iii) the bare game plus “Before picking, briefly predict the distribution of the other players’ picks; then give your pick on the final line.” Bin every pick against the ladder of Exercise 6 — level-0 (40–60), level-1 (28–39), level-2 (18–27), level-3 or deeper (4–17), equilibrium play (0–3), other — and tabulate by regime. Then answer: does regime (ii) move play toward the equilibrium, as it should if the model conditions its depth on its opponents’ presumed rationality; does regime (iii) deepen play relative to (i), and by roughly how many rungs — Section 9.6’s claim that prompting opponent-modelling cranks the k by hand, here measured; and, for each regime, compute the empirical mean of the model’s own picks and where two-thirds of it falls, to determine which regime would win against a field of its own copies. Record the model identifier and the date beside the table: the numbers are perishable, and the durable finding is the shift between the regimes, not the occupancy of any bin.