Read a game tree and reason backward without confusing the analysis with play order.
By the end: apply backward induction to a small observed-choice game and specify off-path actions.
A tree records what can follow what
An entrant chooses Out or In. Out ends the interaction at entrant/incumbent payoffs 0,3. If entry occurs, the incumbent observes it and chooses Fight or Share. Fight gives −1,−1; Share gives 1,1. Edges are actions and terminal pairs are full utilities. This is perfect information because the decision-maker observes the preceding moves.
A strategy specifies an action at every decision point where that player might move, including points not reached in actual play. The incumbent's strategy must say what it would do after entry even if the entrant stays out. An observed outcome alone does not reveal this complete plan.
Read a terminal payoff before reasoning backward
A decision node is a point where a player chooses among outgoing branches. A terminal node ends the interaction and records its full consequences as a utility pair. In the entry tree, the first coordinate always belongs to the entrant and the second to the incumbent. Follow Out and the game ends at 0,3; there is no later response on that path. Follow In and another decision becomes available. Fight and Share are alternatives at that later node, not events that must both occur. Branch lengths and angles are schematic and do not measure time or probability.
Read the baseline tree forward once without solving it. Out gives the entrant 0 and the incumbent 3. In followed by Fight gives negative 1 to each. In followed by Share gives 1 to each. The incumbent's attractive 3 at Out is not an action it can choose after entry. Once its node has been reached, its available utilities are negative 1 and 1. Comparing 3 with these values as though the incumbent could now choose Out would add an unavailable action. A valid comparison starts by identifying the decision-maker and the branches still available there.
Only available branches belong in a node's comparison
Baseline tree, schematic geometry. The incumbent selects between Fight and Share, not the already unavailable Out outcome.
The same restriction applies to the entrant. It cannot select the terminal outcome In, Share by itself. It can select In, after which the incumbent chooses the continuation. Earlier matrix lessons held another player's strategy fixed when testing a deviation. Here the tree makes the intervening choice visible. To evaluate In, the entrant needs an account of the incumbent's later response. The advantage of a tree is that it shows where that response is chosen and what the incumbent will know then. It does not remove the need to justify a prediction about that response.
Perfect information means that at each decision point the player observes all earlier actions in this game. It does not mean unlimited intelligence or knowledge of every fact in the world. This lesson additionally stipulates the displayed utility numbers and analyzes choices as best responses. Those are separate assumptions. An incumbent can observe entry yet have preferences different from the displayed numbers. Conversely, a researcher may know the payoffs but draw the information structure incorrectly. Check observation, available actions, and utility representation separately rather than treating a clear tree as evidence that all three are accurate.
A strategy specifies what a player would choose at every decision point it might face. For the entrant here, that is simply Out or In. For the incumbent, it is Fight after entry or Share after entry, even if entry never occurs. A realized path is only the sequence of actions actually taken. With Out, the realized path contains no incumbent action, but the incumbent's strategy still matters for the entrant's comparison. This distinction prevents us from mistaking an untested plan for an observed response, or from dropping that plan merely because it remains unreached.
Response plans remain distinct even when Out makes them unreached
Entrant strategy / incumbent plan
Fight after In
Share after In
Out
0, 3 · incumbent indifferent across plans
0, 3 · entrant can gain by entering
In
−1, −1 · both can gain by switching
1, 1 · both best responding
Paper representation of the baseline tree. Columns specify complete response plans, not observed incumbent actions.
As a paper representation of the same game, write a matrix whose columns are the incumbent's response plans. Both cells in the Out row contain 0,3 because neither response is reached. The In row contains negative 1,negative 1 under Fight and 1,1 under Share. This matrix is not another game preset in the explorer. It helps connect tree strategies to the earlier best-response method. The identical Out payoffs do not make the response plans identical strategies: the In row reveals their different consequences when the earlier choice changes.
Work backward, play forward
Start at the incumbent's decision. After entry, Share gives the incumbent 1 instead of −1, so Share is optimal. Carry that continuation payoff back to the entrant. In then gives 1 instead of Out's 0, so entry follows. The analysis runs backward; actual play still begins with the entrant.
The player resolves the final choice and then the initial choice. Changing the Fight payoff can change the continuation and therefore the earlier action. The example selects Share when the incumbent is indifferent and Out when the entrant is indifferent. These tie conventions select one solution rather than asserting uniqueness.
Carry the whole continuation outcome back to the earlier choice
Backward induction starts with choices that have no unresolved decisions after them. Resolve each using the utility of the player who moves there, then use the selected continuation when analyzing an earlier node. This is reasoning order, not play order. In the baseline tree, start after entry. The incumbent compares its own negative 1 from Fight with its own 1 from Share and selects Share. This comparison does not use the entrant's payoff. Agreement between the players' rankings in this baseline case is incidental to the method, not a requirement for applying it.
Carry the entire selected pair 1,1 back to the In branch. The entrant now compares its own 1 from that continuation with its own 0 from Out and chooses In. Do not carry only the incumbent's utility back and then let the entrant maximize it. Keeping the pair prevents that coordinate error. The resulting complete profile is In for the entrant and Share if entry occurs for the incumbent. Actual play begins with In and then proceeds to Share. The backward calculation explains that forward sequence without making either player move backward in time.
Resolve the later comparison before the earlier one
Scenario
Incumbent compares Fight / Share
Pair carried to In
Entrant compares In / Out
Selected profile
Fight utility −1
−1 / 1: Share
1, 1
1 / 0: In
In; Share after In
Fight utility 2
2 / 1: Fight
−1, 2
−1 / 0: Out
Out; Fight after In
Carry the selected pair backward, then read the earlier player's coordinate. Utilities belong to different decision-makers.
Next change only the incumbent's Fight utility to 2, as in the deterrence preset. The entrant's Fight utility remains negative 1, and Share still gives 1,1. The incumbent now prefers Fight because 2 exceeds 1. The selected continuation pair is therefore negative 1,2. Returning to the root, the entrant prefers Out's 0 to In's negative 1. The selected outcome becomes 0,3, but the full profile still includes Fight after entry. Deterrence here means that anticipated opposition makes entry unattractive, so the response remains hypothetical under the selected strategies.
This comparison distinguishes a change in incentives from a change in language. Increasing the incumbent's Fight utility changes what it prefers when entry has already occurred. Announcing Fight without changing the baseline utilities would leave Share preferable there. The later commitment lesson examines a different way to change continuation incentives through an enforced penalty. No such penalty appears in this lesson's three presets. Here the scenario change directly stipulates a different Fight utility. The payoff parameter is an input to the teaching example, not a measurement of aggressiveness or a conclusion drawn from an announcement.
At the third preset, Fight gives the incumbent 1, exactly equal to Share. Both responses maximize incumbent utility. The implementation selects Share at this tie, so the entrant anticipates 1 and chooses In. Another allowed tie selection could choose Fight and support Out instead. Thus the selected answer does not establish a unique solution for the underlying tied game. The discontinuity is informative: raising Fight utility from just below 1 to just above 1 can change the selected earlier action even though the entrant's own terminal payoffs have not changed at all.
A later tie can support different earlier comparisons
Incumbent plan after In
Incumbent utility
Entrant utility from In
Entrant best response
Share for certain
1
1
In
Fight for certain
1
−1
Out
Half Fight, half Share
1
0
In or Out
Paper alternatives at Fight utility 1. The lab selects Share; it does not offer a response-mixture control.
For a separate paper calculation within the same tied game, suppose the incumbent randomly Fights half the time after entry. Its expected utility remains 1 because both responses give 1. The entrant's utility from In becomes one half times negative 1 plus one half times 1, or zero. The entrant is now indifferent between In and Out. The lab does not provide a response-probability control; it selects Share at the incumbent tie and Out at any entrant tie. This paper mixture shows why declaring a tie convention is different from proving that alternatives are impossible.
Compare incentives and outcomes
Resolve the response, then the initial choice
Event 0 of 2. All terminal outcomes shown.
Step 0 / 2
Complete continuation calculation
Decision
Own alternatives
Selected
Incumbent after In
Fight -1; Share 1
Share
Entrant
Out 0; In 1
In
Changing the Fight payoff resets playback. Reset retains that payoff. Event one resolves the incumbent response; event two resolves the entrant choice. This is reasoning order, not play order. Share wins incumbent ties; Out wins entrant ties. Blue thick branches identify selected actions.
Analysis step
Comparison
Choice
Incumbent after In
1 versus −1
Share
Entrant anticipating Share
1 versus 0
In
Continuation incentives determine the earlier comparison.
Use playback to inspect the two comparisons separately
Select Fight costs: negative 1 and inspect the full continuation calculation as well as the drawing. The shared player has two reasoning events. The first selects the incumbent response after In; the second selects the entrant's earlier action. Terminal utilities remain visible before either event. Under reduced motion, manual stepping exposes these events without requiring moving branches. Reset retains the chosen Fight scenario and returns the reasoning to its start. Changing scenario resets playback to analyze a different game. These controls change the explanation's stage, not the order in which the fictional players would act.
At the first event in the baseline case, Share should receive the thick blue branch because the incumbent compares 1 with negative 1. Before looking at the second event, evaluate the entrant's two utilities using that selected response. Out gives 0; In now leads to 1. The second event therefore adds the selected In branch. This is a useful check against reading the attractive incumbent payoff of 3 as a command to stay out. That 3 belongs to the other player and does not determine the entrant's own preferred choice.
Out and a hypothetical Fight can both belong to the selected plan
Thick branches specify strategies. Out is realized; Fight is the optimal response only if entry occurs.
Switch to Fight pays: 2 and repeat the two comparisons. First locate the incumbent's coordinate in negative 1,2 and compare 2 with Share's 1. Then carry the selected pair back and compare the entrant's negative 1 with Out's 0. At completion the drawing emphasizes Out and the Fight response. These thick branches cannot both lie on the realized path, since Out ends play before Fight. They represent a complete selected contingent plan, including what would happen after hypothetical entry. Reading them as a recording of two actions would contradict the tree's structure.
Switch to Fight ties Share: 1 and inspect the equality rather than rounding it into a strict preference. The selected Share branch comes from the stated incumbent tie convention. The entrant's subsequent preference for In is strict because 1 exceeds 0. Open the complete calculation if playback has already finished; it preserves both comparisons without requiring you to catch a transient highlight. The three scenarios isolate the effect of one changed incumbent payoff. They do not let you edit entry costs, alter the information available, choose arbitrary response rules, or replay a sample of actual behavior.
A separate paper stress test changes the entrant's Out utility from 0 to 2 while keeping the baseline continuation Share at 1,1. Backward reasoning still selects Share after entry because the incumbent's comparison is unchanged. The entrant now chooses Out because 2 exceeds 1. Thus observing Out would not by itself distinguish a profitable outside alternative from anticipated fighting. In both cases no response occurs on the realized path, yet the continuation strategies differ. This paper test is not an available control; it identifies which missing payoff evidence would matter in interpreting an observed outcome.
An outside alternative can change entry without changing the response
Entrant Out utility
Incumbent selected response
Entrant In utility
Entrant optimal action
0, actual baseline
Share
1
In strictly
1, paper tie
Share
1
In or Out; convention selects Out
2, paper alternative
Share
1
Out strictly
Paper changes, not controls. Share remains optimal after entry; only the entrant's root comparison changes.
Another paper stress test sets the entrant's Out utility to 1. With baseline Share still selected, both root branches give the entrant 1. The implementation's stated entrant tie convention would select Out in that hypothetical comparison, but both choices are best responses in the game itself. Small changes to an outside alternative can therefore separate a strict conclusion from a tie-selected one. To transfer the method, record the later best-response comparison and the earlier comparison separately. If one input changes, recompute the affected comparison rather than assuming that every selected branch must change.
Check every subgame
Subgame perfection requires a Nash equilibrium in every subgame, including unreached ones. In a finite perfect-information game, backward induction supplies such a strategy profile. With strict choices at every node, the recursive solution is unambiguous. Ties can create multiple solutions.
Do not apply this procedure blindly to hidden actions. If two decision nodes belong to the same information set, the player cannot condition a choice on which node they occupy. Nor does a solution establish that real people correctly know the payoffs or reason through the entire tree.
Check unreached decisions without granting hidden observations
A subgame is a continuation that begins at a decision point and includes everything that can follow it, without splitting a player's information set. An information set groups decision points the player cannot distinguish when choosing. In this entry example, the incumbent knows entry occurred and has just one possible response node. The continuation beginning there is therefore a subgame. Subgame perfection requires the strategies to form a Nash equilibrium in that continuation as well as in the whole game. It checks whether a selected response remains optimal when its decision point is actually reached.
The baseline profile Out together with Fight after entry passes the whole-game Nash test. Against a fixed Fight plan, the entrant prefers Out's 0 to negative 1. Against certain Out, changing the incumbent's unreached response leaves its utility at 3, so it cannot improve. Yet in the continuation after entry, Share gives the incumbent 1 instead of negative 1. Fight fails that local comparison. This profile is Nash but not subgame perfect. The next lesson develops this credibility issue; here it explains why checking a whole-game matrix need not reproduce backward induction.
An unreached strategy can pass Nash and fail the continuation test
Out plus Fight plan
Entrant: Out / deviation In
Incumbent: current / changed plan
After entry: Fight / Share
Subgame perfect?
Baseline Fight utility −1
0 / −1
3 / 3
−1 / 1
No
Fight utility 2
0 / −1
3 / 3
2 / 1
Yes
The same Out profile passes whole-game deviations in both scenarios, but only the second Fight plan is credible.
With Fight utility 2, the profile Out together with Fight passes both tests. The whole-game deviations still fail, and after entry Fight now gives the incumbent 2 rather than Share's 1. The unreached plan is optimal if called upon. This does not mean fighting is good for the entrant or socially desirable. It means that the specified incumbent preference supports the plan at the relevant node. Subgame perfection adds a consistency requirement to the strategic explanation. It does not add a fairness criterion or convert the displayed utility pair into a common welfare scale.
Information can prevent the local comparisons used in backward induction. Consider a separate paper version of Match or differ where Row chooses first but Column cannot observe that action. Draw a Column node after each possible Row action and join those nodes into one information set. Column must use the same choice rule at both nodes because it cannot tell which it occupies. Solving each node independently would wrongly let Column choose Right after Left and Left after Right. That would guarantee a mismatch by supplying an observation that the paper game explicitly withholds.
If Column instead observes Row's action in that paper game, those two response nodes become distinguishable. Column can now choose Right after Left and Left after Right, obtaining 1 in either case while Row gets negative 1. Row cannot avoid that outcome by choosing a different pure action or privately mixing, because Column responds after observing the realized action. Under hidden observation, independent half-half mixing instead yields zero to both at equilibrium. This side-by-side comparison changes only observation, not the order on the clock. It explains why physically moving second is not sufficient for a response advantage.
Observation changes which response plans are feasible
Paper Match or differ trees, schematic geometry. The dotted information set forbids conditioning on Row's hidden action.
The linked Yale lectures on strategies and information sets explain complete contingent strategies, backward reasoning, and the distinction between information and timing. Their examples are teaching models, as are the entry payoffs used here. Applying the method to an institution requires evidence that later players can observe the relevant action, have the stated alternatives, and rank the consequences as assumed. An observed Out path supplies none of the missing off-path choices by itself. A defensible conclusion names the continuation incentives that support a profile and the information needed to condition on earlier actions, rather than treating a solved drawing as an empirical forecast.
Exercises
Reverse the continuation incentive
Keep all terminal payoffs except Fight, which now gives −1,2. What does backward induction select?
Hint
The incumbent compares its own 2 with its own 1.
Worked answer
Decision
Own payoff comparison
Choice
Incumbent after In
2 > 1
Fight
Entrant
0 > −1
Out
The incumbent's off-path action is part of the solution.
The incumbent chooses Fight after entry. Anticipating entrant utility −1 rather than Out's 0, the entrant stays out. The full strategy still includes Fight after hypothetical entry, even though that branch is not reached.
Change information, not the clock
Column physically moves after Row but cannot observe Row's action. Can Column choose Left after Row's Left and Right after Row's Right?
Hint
A strategy can depend only on available information.
Worked answer
What Column knows
Permitted plan
Row's actual move is hidden
One choice rule at the information set
Row's move is observed
A separate response to each observed move
Timing and information are distinct model inputs.
Not in this information structure. The two decision nodes are indistinguishable to Column and must share the same action or randomized rule. Merely moving later does not create an observation. Drawing separate visible-choice branches and solving each independently would give Column information they do not possess.
Sources and further reading
Selected reference passages support the definitions and algorithms below. The games, explanations, and implementations are original. Terminology and assumptions are stated within each lesson; source review is not an independent audit of this course. Optional links are not runtime dependencies.
Selected transcript on solving continuation decisions and how enforceable collateral changes incentives. No actual loan or enforcement terms are inferred.