Represent private types and compare decisions with and without additional information.
By the end: calculate a type-weighted best response without granting a player unobserved knowledge.
A type describes private information
Suppose the incumbent knows its own type. A Tough type prefers Fight after entry, with incumbent utilities 3 for Fight and 1 for Share. A Flexible type prefers Share, with −1 for Fight and 1 for Share. The entrant does not observe the type. In this separate example, entry gives the entrant −2 against Tough and 2 against Flexible; Out gives 0.
A Bayesian game specifies types, a distribution or prior belief, what each player knows, and type-dependent payoffs. A strategy specifies a choice for each type or information situation the player can distinguish. Knowing that there are two types does not let the entrant condition its decision on the actual hidden type.
A hidden type restricts the entrant's feasible plan
A type collects facts about a player that matter to the game but are not all publicly observed. Here Tough and Flexible describe the incumbent's preferences after entry. They are not personality diagnoses. Tough gets 3 from Fight and 1 from Share, so Fight is its strict best response. Flexible gets minus 1 from Fight and 1 from Share, so Share is its strict best response. Each type knows which comparison applies to itself. The entrant knows these comparisons but does not know which type it faces.
We can represent the uncertainty by putting a chance choice before the entrant's decision. Chance is a mathematical device for assigning probabilities, not a third person with goals. It selects Tough with probability p and Flexible with probability one minus p. An information set groups decision points that a player cannot distinguish when choosing. The entrant's two possible decision points belong to one information set because it has received no type observation. Drawing separate branches on a page does not grant the entrant separate observations in the game.
Two hidden states do not create two informed choices
The entrant cannot distinguish its linked nodes. Geometry is schematic; chance probabilities and terminal entrant utilities are explicit.
The restriction changes what counts as a strategy. Without disclosure, a pure strategy is Enter in both states or Out in both states. With reliable disclosure before the decision, a strategy can say Out against Tough and Enter against Flexible. That conditional plan is unavailable in the hidden-type game. A randomized uninformed strategy is possible, but its entry probability must be the same against both types. Otherwise its randomizer would somehow have to read the hidden type. The earlier distinction between a complete strategy and a realized action still applies.
Use a paper population of one hundred possible encounters to make the information restriction concrete. At p equal to 25%, twenty-five encounters have Tough incumbents and seventy-five have Flexible incumbents. These counts represent the prior probabilities exactly for calculation; the lab does not generate encounters. Always Enter earns twenty-five times minus 2 plus seventy-five times 2, a total of 100 utility units across the paper cases. Dividing by one hundred gives expected utility 1. It does not mean any single encounter pays 1. Each entry pays either minus 2 or 2.
A plan that enters only in the seventy-five Flexible cases earns 150 across those cases and zero in the others, averaging 1.5. Its advantage comes from conditioning the action on a fact, not from changing either entry payoff. Compare that with a random rule that enters in seventy-five percent of all cases independently of type. That rule still sometimes enters against Tough and sometimes misses Flexible. Its expected utility is three quarters of the always-Enter value, or 0.75. Matching the overall frequency of entry does not reproduce an informed strategy.
Matching an entry rate does not match informed selection
Rule
Tough contribution
Flexible contribution
Expected utility
Always Enter
0.25 × (−2) = −0.5
0.75 × 2 = 1.5
1
Enter independently with probability 0.75
0.75 × (−0.5) = −0.375
0.75 × 1.5 = 1.125
0.75
Enter only if Flexible
0
0.75 × 2 = 1.5
1.5
Paper cases at a 25% Tough prior. Independent entry cannot direct its decisions toward the Flexible cases.
The probability p is a prior, meaning the belief used before receiving any additional evidence in this example. It need not equal a measured fraction of incumbents in a city or industry. If the claim concerns one particular opponent, evidence about that opponent may matter more than a broad population frequency. The arithmetic takes p as supplied and asks which decision follows. It does not estimate the prior, test whether the labels fit real people, or establish that an incumbent's preferences stay fixed across different circumstances.
Average over the uncertainty you actually have
Let p be the entrant's probability that the incumbent is Tough. Entering has expected utility 2 minus 4p. Out has utility 0. Enter is strictly better below p = one half, Out above it, and both are best responses at equality. The incumbent's stated responses are optimal for each type.
A Bayesian Nash equilibrium requires each type's strategy to be a best response under the specified beliefs and other strategies. Here the explorer checks a simple entry decision against stipulated, type-optimal responses. It is not a general equilibrium solver and does not infer p from an observed action.
Derive the entry threshold and the best uninformed value
Expected utility is a probability-weighted average of the utilities available to the same decision-maker. To calculate entry, multiply its Tough payoff, minus 2, by p. Then multiply its Flexible payoff, 2, by one minus p. Add these contributions. The Flexible contribution is 2 minus 2p, and the Tough contribution subtracts another 2p, leaving 2 minus 4p. Out gives zero in either state, so its weighted average is zero. We compare those two averages, rather than comparing one favorable entry outcome with the guaranteed outside outcome.
The threshold is the prior at which these averages tie. Entry starts at 2 when Tough has probability zero. Each additional percentage point of Tough replaces a one-percent chance of receiving 2 with a one-percent chance of receiving minus 2. That lowers expected utility by 0.04. Fifty percentage points therefore remove the initial 2 units. At p equal to one half, entry gives zero. Below one half it is positive; above one half it is negative. This reasoning obtains the threshold with arithmetic and also explains the slope of the entry line.
The best uninformed action switches at one half
The declining line averages entry utilities. The better of that line and Out determines the best uninformed value.
At p equal to 40%, the Tough contribution is minus 0.8 and the Flexible contribution is 1.2, leaving 0.4. At 60%, the contributions are minus 1.2 and 0.8, leaving minus 0.4. The best uninformed value is therefore 0.4 in the first case and zero in the second, because Out remains available. This maximum of two available utilities is different from the entry utility alone. Reporting minus 0.4 as the decision-maker's best available value would silently remove the option of declining entry.
Could randomization improve that best value? Let r be the probability of Enter, chosen without observing type. Since Out pays zero, the rule's expected utility is r times the entry utility. At p equal to 40%, choosing r equal to one half gives 0.2, less than always entering. At 60%, the same rule gives minus 0.2, less than staying out. At 50%, every r gives zero. Randomization cannot exceed the better pure action here. It creates more best responses at the tie, not a strict improvement over both actions.
Notice why this differs from mixed equilibrium in matching pennies. There, each player's mixture affects the other player's incentive to deviate. Here the calculation holds fixed the probability of types and the incumbent's type-optimal responses. The entrant is selecting a best response to those inputs. A complete Bayesian equilibrium would require every type's strategy and each relevant belief to fit together. In this example the incumbent checks are simple because both response comparisons are strict, but the lab's expected-value readout is not a general method for solving arbitrary games with private information.
An uninformed mixture cannot beat both pure actions
Prior Tough
Always Out, r = 0
Half entry, r = 0.5
Always Enter, r = 1
40%
0
0.2
0.4
50%
0
0
0
60%
0
−0.2
−0.4
Entry probability r applies to both types. At the equal prior, all these rules tie at zero expected utility.
The payoffs in this lesson are also different from those in the earlier observed-entry tree. Here entering against Tough gives the entrant minus 2, while entering against Flexible gives 2. Importing the earlier minus 1 and 1 outcomes would preserve the one-half threshold only because both magnitudes change proportionally. It would halve every entry utility and every information value. A familiar-looking threshold can therefore hide a wrong calculation. Keep the terminal utilities attached to their own example before comparing models or interpreting a numerical result.
Compare incentives and outcomes
Compare uninformed and informed choices
Type hidden
Action
Expected utility
Enter
0
Out
0
Best available: 0.
Type revealed first
Observed type
Action
Utility
Tough
Out
0
Flexible
Enter
2
Expected utility: 1.
Value of perfect information: 1. Uninformed actions tie.
Average entry across the two types
p is the probability of Tough. The −2 and +2 are entrant utilities in this example, not those in the earlier entry tree.
Presets replace only the prior probability of Tough. Conditional payoffs and type-optimal incumbent responses stay fixed. Perfect information arrives before entry and does not change behavior or payoffs. Values are expected utility, not a forecast of a particular outcome.
Hidden type
Probability at p = 1/2
Entry utility
Informed action
Tough
1/2
−2
Out
Flexible
1/2
2
Enter
The informed plan can condition on type; the uninformed plan cannot.
Use the prior control to isolate the value of disclosure
In the lab, select Mostly Flexible: 25%. Before reading the information value, calculate the best uninformed choice. Entry yields 1, so the uninformed entrant already chooses Enter. The revealed-type plan receives 2 in the Flexible cases, which have probability 0.75, and zero otherwise. Its value is 1.5. The improvement is 0.5, not 1.5. The smaller number measures what disclosure adds to an already competent decision. Confusing the two values would count the profitable entries that were already available without buying any information.
Now select Mostly Tough: 75%. The entry average becomes minus 1, so the best uninformed choice changes to Out, with value zero. Reliable disclosure still permits entering in the Flexible quarter of cases, yielding 0.5. Its improvement is again 0.5. The symmetry has a different explanation on each side of the threshold. At 25%, information prevents occasional harmful entries. At 75%, it permits occasional beneficial entries. The same information value can reflect avoided losses or newly available gains, depending on the best uninformed action being replaced.
Disclosure gains are symmetric around the threshold
Prior Tough
Entry
Best uninformed
Informed
Information gain
0%
2
2
2
0
25%
1
1
1.5
0.5
50%
0
0
1
1
75%
−1
0
0.5
0.5
100%
−2
0
0
0
The gain subtracts the best uninformed value, which switches actions at one half. Negative entry values are not baselines.
Next select Equal prior: 50%. Entry and Out both give zero before disclosure. After disclosure, the entrant enters in the Flexible half and receives 2 there, giving expected utility 1. This is the largest information value in this particular model. Move the prior slider toward either endpoint and track both panels, not just the entry average. At zero, entering is already always correct. At one hundred percent Tough, staying out is already always correct. Disclosure adds nothing at either endpoint, even though the two best uninformed actions are opposite.
We can derive the entire information-value pattern without inspecting every slider position. When p is at most one half, the best uninformed value is 2 minus 4p. Subtract it from informed value 2 minus 2p to obtain 2p. When p is at least one half, the best uninformed value is zero, so information adds 2 minus 2p. Both calculations give 1 at the boundary. This produces a triangular value curve, while the total informed utility is a straight declining line. They answer different questions and should not be given the same label.
As a separate paper purchase calculation, suppose perfectly reliable disclosure costs 0.6 utility units regardless of which type appears. At p equal to 25%, informed utility after this cost is 1.5 minus 0.6, or 0.9. That is worse than uninformed entry's 1. At 50%, informed utility after cost is 1 minus 0.6, or 0.4, better than zero. There is no disclosure-price control in the lab. This extension assumes the cost can be subtracted on the entrant's utility scale and does not change its other options or the incumbent's incentives.
Information gain is not total informed utility
Exact scales show a straight informed-value line and a triangular incremental-value line. Both reach zero at certainty of Tough.
Changing the prior control replaces only the Tough probability. It does not reveal a sampled incumbent, collect observations, or update the prior after an action. The conditional payoff and response assumptions remain fixed. This makes the control useful for asking how sensitive a decision is to an uncertain belief. For example, moving between 49% and 51% changes the strict best response, although expected entry utility changes only from 0.04 to minus 0.04. A sharp action threshold does not imply a large utility difference near that threshold.
Value information before paying for it
If a perfectly reliable disclosure reveals the type before entry, the entrant can stay out against Tough and enter against Flexible. Expected utility becomes 2 times 1 minus p. Compare this with the best uninformed action, not with an arbitrarily bad uninformed action.
At p = one half, the uninformed value is 0 and the informed value is 1. Information worth at most 1 utility unit is not worth a price of 1.5 in this model. The value assumes disclosure does not change the incumbent's incentives, available actions, or payoffs. A strategic message is not automatically such a disclosure.
Separate reliable evidence from a strategic message
Perfect disclosure is valuable here because the entrant may use the information and may also ignore it. Its expanded set of decision rules contains both old rules, always Enter and always Out. It also contains the selective rule that avoids Tough while entering against Flexible. Choosing the best rule from this larger set cannot lower expected utility before any information fee. That argument depends on the information arriving before the action and leaving payoffs and available actions unchanged. A report delivered after irreversible entry cannot support the selective entry plan calculated here.
For a paper extension, consider an imperfect report that says favorable or unfavorable. Use one hundred equally weighted cases with an equal prior. Suppose the report is favorable in forty Flexible cases and ten Tough cases, and unfavorable in ten Flexible cases and forty Tough cases. Among the fifty favorable reports, Tough accounts for ten, so the conditional Tough probability is 20%. A conditional probability is a probability calculated within the cases compatible with the evidence. The report changes which cases belong in the entrant's average; it does not change anyone's actual type.
Condition on the report before averaging utilities
Invented paper report, not a lab control. The same prior counts yield different conditional probabilities within the two report groups.
After a favorable report, entry gives 2 minus four times 0.2, or 1.2. After an unfavorable report, the Tough probability is 80%, so entry gives minus 1.2 and Out is better. Entering only after favorable reports produces forty gains of 2 and ten losses of 2, totaling 60 across one hundred cases. Its expected utility is 0.6. This falls between the uninformed best value zero and perfect-disclosure value 1. These counts define an invented report technology, not a feature of the slider or an empirical accuracy estimate.
A message chosen by the incumbent requires a different calculation. If both types want to discourage entry, merely saying 'I am Tough' may be attractive to both. Hearing the same message then does not separate them. To interpret the message, ask how often each type would choose it and why. A statement whose truth can be independently verified has different incentives from an unsupported assertion. Even silence can carry information if the opportunity to disclose is known and disclosure choices differ by type. None of those message strategies is supplied by the perfect-information benchmark.
The next lesson examines a certificate whose cost differs across types. That is a way to make the proposed communication strategy survive an imitation check. It is not the same as giving the entrant a free observation of a hidden fact. Here we supplied a probability and held type responses fixed. In a signaling problem, the probability attached to an observation must also reflect who chooses to produce it. Taking the meaning of a signal for granted can therefore produce correct arithmetic about the wrong conditional probabilities and a mistaken behavioral prediction.
A selective report rule earns less than perfect disclosure
Policy across 100 cases
Flexible entry gains
Tough entry losses
Expected utility
Enter after either report
50 × 2 = 100
50 × (−2) = −100
0
Enter only after favorable
40 × 2 = 80
10 × (−2) = −20
0.6
Enter only when type is Flexible
50 × 2 = 100
0
1
Paper cases from the report diagram. The favorable-only policy avoids most Tough entries but still misses ten profitable Flexible entries.
For transfer to another decision, first identify what remains unknown when the action must be selected. Write the available action's utility in each hidden state, calculate its average, and retain the best uninformed alternative. Then ask which additional conditional rules the proposed evidence actually makes feasible. Value the improvement over that baseline and subtract any justified utility cost. If evidence changes the opponent's behavior or the feasible actions, revise those inputs too. The arithmetic can compare specified information arrangements; it cannot by itself establish their reliability, credibility, or desirability for other people.
Exercises
Price a perfect disclosure
At p = 0.25, what is the best uninformed utility, the perfectly informed utility, and the maximum willingness to pay for the disclosure?
Hint
The uninformed entrant already benefits from entering.
Worked answer
Plan
Expected utility
Always enter
1
Stay out
0
Enter only if Flexible
1.5
Information gain over best uninformed
0.5
Value information relative to the best available baseline.
Uninformed entry gives 2 − 4 × 0.25 = 1, better than Out's 0. Perfect information gives 0.75 × 2 = 1.5. Its incremental value is 0.5, not 1.5. A price below 0.5 strictly improves utility; equality leaves indifference.
Private information is not a free observation
The entrant says, "I will enter if the incumbent is Flexible and stay out if Tough," but has no type signal. Is that a feasible strategy?
Hint
Ask what the entrant can distinguish before choosing.
Worked answer
Information
Feasible decision rule
No type observation
One action or mixture across both states
Type observed
Separate choices for Tough and Flexible
Do not solve an easier information problem than the one specified.
No. It is a contingent plan requiring information the entrant does not possess. It becomes feasible after a reliable type observation, not merely after writing two branches in a diagram. An uninformed strategy must use the same decision rule across those indistinguishable states.
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 candidate-separation incentive checks and the role of different signal costs. Our constructed certificate is not empirical evidence about education or hiring.