Complete plans and information sets

What you will learn

Construct a complete pure strategy that is constant within each information set, count admissible plans and distinguish feasible response value from a clairvoyant benchmark.

Prerequisites: Information and incentives · Credible continuations and commitment

A strategy is more than one realized action

A complete pure strategy specifies what a player would do at every information set where they might act. An information set groups decision histories the player cannot distinguish from their observations. A plan cannot choose different actions at histories that look identical to that player. The model editor can display all histories without giving the in-game player access to their hidden labels.

This requirement includes contingencies that are not reached in a particular play. A realized route reveals one action and one outcome; it does not describe the whole strategy. Counting actions at individual hidden histories can overstate the player’s actual choices. Complete plans and realized paths are different objects.

Construct the observation partition first

An opponent follows a supplied fixed policy: choose one of four routes A,B,C,Dwith probability1/4each. The learner then chooses XorYafter receiving a signal. Route receipts(X,Y)are A(6,0), B(0,2), C(2,0), D(0,6). These artificial receipts and the route law stay unchanged while observation rights vary. The treasury pays receipts; this is not a claim of zero-sum strategic equilibrium.

With no route information, all four histories share one signal. Only XXXXandYYYYare legal pure plans, each with mean receipt2. A different X/Ychoice at each hidden route would use information the player was not given, even though the editor displays each payoff case. Observing the route after fixing the plan cannot retroactively change its action.

A coarse signal permits some contingencies, not all

Now signal S0means AorB, while S1means CorD. The action at Amust equal the action at B; the action at Cmust equal the action at D. With two available actions at each of two information sets, there are2²=4legal pure plans. In general this product is2^kfor ksets when both actions are available at every set.

Inside S0, Xreceipts sum6and Yreceipts sum2, so choose X. Inside S1, Xsums2and Ysums6, so choose Y. The resulting complete plan XXYYhas route receipts6,0,0,6and mean3. Choosing XYXYwould earn4if the route were known, but it is not a legal plan under this coarse observation map.

State the benchmark’s information and opponent assumptions

With full route observation, four singleton information sets allow2⁴=16pure plans. XYXYthen becomes feasible, earning receipts6,2,2,6and mean4. This is the full-information benchmark. The feasible gaps are2with no information,1with the coarse signal and0with the full route. Finer costless observation cannot reduce the best response value against the same fixed opponent law, because the learner can still ignore the extra signal.

That is not a universal strategic claim that more information always helps. An opponent who changes behavior after learning your observation rights changes the comparison. Nor does a best response to a supplied policy establish equilibrium, exploitability against every opponent or full-poker solver fidelity. The activity fixes the route law and tests only the specified finite decision problem.

Hands-on: Build information sets and program a complete strategy

Link indistinguishable histories, test distinct legal plans against a fixed opponent policy, choose a best feasible plan and reconcile every route ledger and information benchmark.

Reading complete-plan workshop…

Common mistakes

• Treating the editor’s route label as an in-game observation.
• Choosing different actions at histories with the same signal.
• Confusing a realized path with a complete contingent plan.
• Counting2⁴plans when only two information sets are available.
• Calling a route-wise benchmark feasible before the route is known.
• Assuming the opponent adapts when the comparison explicitly holds its policy fixed.

Keep these ideas

• Build information sets from observations, then define actions on those sets.
• A complete pure strategy covers all relevant contingencies.
• With two actions at each of ksets, there are2^klegal pure plans.
• Benchmark comparisons must specify information rights and a common opponent law.

Further reading

• Kate Larson — CS886 Extensive Form Games, strategies and information sets ↗

  Bounded official course-slide review, PDF pages14,29and33: complete contingent strategies and one action per information set. Original four-route observation maps, receipt tables and artwork; no source exercise, whole slide-deck review, equilibrium solution or video claim. Checked 2026-10-22.

• Ben Polak — ECON159 Lecture14, backward induction and commitment (Open Yale Courses) ↗

  Bounded official transcript review of backward reasoning and the commitment/observability discussion. Our discrete enforced-bond game, fake-chip treasury and drawings are original; no Stackelberg derivation, lecture exercise or full-video review claimed. Checked 2026-10-22.