Use outside options and an explicit offer protocol to reason about agreement.
By the end: identify mutually acceptable splits and distinguish those from a predicted or fair bargain.
Start with the disagreement outcome
Two parties can divide 12 units if they agree. Let A's outside option be a and B's be b. These are utilities if they do not agree, not claims on the same 12 units. An offer of x to B leaves 12 minus x for A. Both weakly prefer it to disagreement only when x is at least b and at most 12 minus a.
With outside options 2 and 4, B's feasible offer interval is 4 through 10. Total gains over disagreement are 12 − 2 − 4 = 6. This calculation identifies available surplus. It does not identify a unique split, prove the outside options are credible, or decide what distribution is just.
Find each party's gain over walking away
An outside option is the outcome a party can obtain if this agreement fails. It is a comparison point, not an amount removed from the proposed division. With A's option at 2 and B's at 4, the twelve units remain available for agreement. Offering B 6 leaves A 6. A gains 4 above walking away, while B gains 2. The equal division of the twelve units therefore gives unequal gains over disagreement. This distinction matters whenever the parties start with different alternatives, even if the agreement itself looks symmetric.
To locate all acceptable offers, begin with B rather than a proposed compromise. B needs at least 4, so offers 0 through 3 fail B's comparison. Now check A. A needs to retain at least 2, so B can receive at most 10. Intersect the two requirements, meaning keep only offers satisfying both. The resulting integer offers are 4, 5, 6, 7, 8, 9, and 10. There are seven offers because both endpoints count. Feasibility identifies a set of possibilities before any bargaining rule selects among them.
The same surplus can be divided differently
Offer to B
A keeps
A gain over 2
B gain over 4
4
8
6
0
6
6
4
2
7
5
3
3
10
2
0
6
With outside utilities 2 and 4, each accepted division distributes six additional units. Equal shares need not give equal gains.
The surplus is the additional amount available above the two disagreement requirements. Here the accounting starts with twelve, subtracts A's required 2, then subtracts B's required 4, leaving 6. At an offer of 4, A keeps 8 and gains 6 while B gains zero. At an offer of 10, A gains zero while B gains 6. Intermediate offers distribute those six gains differently. This calculation assumes each added unit gives one additional unit of utility. It does not justify adding unrelated people's reported utility scores in a real negotiation.
Zero surplus and negative surplus have different implications. In a paper example, not an outside-option preset, let A require 7 and B require 5. Only an offer of 5 works, leaving 7 for A. Both are indifferent between agreement and disagreement, so agreement depends on accepting ties. In the available 7,6 preset, B requires at least 6 but A allows at most 5. No offer works. The problem is not failure to find the midpoint. The two requirements cannot fit within twelve under the specified utility assumptions.
A credible outside option must actually remain available when agreement fails. Suppose a buyer says another supplier offers utility 4, but that supplier's offer expires before this negotiation ends. Treating 4 as guaranteed disagreement utility would misstate the game. A costly trip, a delayed payment, or an uncertain alternative also needs evaluation on the same basis as the proposed agreement. Those possibilities are paper considerations, not editable features of this explorer. The lesson's three outside-option presets take the displayed alternatives as certain and immediately comparable with the division.
A single acceptable offer differs from none
Each axis uses the same integer scale. The paper 7,5 alternatives touch at five; the available 7,6 preset has a gap.
Outside options do not establish entitlement. A stronger alternative can raise the minimum someone will accept without showing that their claim is fair. Likewise, someone with no attractive alternative may accept a small amount without endorsing the institutions that created that situation. Keep two questions separate when reading a bargaining claim. Ask which allocations each party prefers to disagreement, then ask which allocations satisfy a stated ethical or legal standard. The first can be answered here by arithmetic. The second needs a standard that the twelve-unit model does not provide.
A protocol supplies additional incentives
In a one-shot take-it-or-leave-it game, A proposes and B accepts or rejects. With money-only utilities, complete information, integer offers, and acceptance at indifference, A offers B the minimum b when agreement also meets A's option. The specified tie rule matters at zero or exactly b.
Real responders may value fairness or reject a low offer for other reasons. Those preferences belong in the model rather than being dismissed as mistakes. Alternating offers, delay costs, deadlines, and uncertain outside options produce different games. Bargaining power cannot be deduced from the size of the pie alone.
Derive an offer from the response rule
A protocol specifies who may make an offer, what responses are available, and what follows a rejection. Consider the stated one-shot protocol with A proposing and B accepting or rejecting. Work backward from B's response, as in the earlier entry tree. With B's outside utility 4, an offer of 3 leads to rejection, 4 leaves indifference, and 5 gives a strict reason to accept. The assumed acceptance rule selects acceptance at 4. A must anticipate this complete response rule, rather than infer it from whether one particular offer was accepted.
Next compare A's possible offers using that response rule. An accepted offer of 4 leaves A 8. Offering 5 leaves A 7, offering 6 leaves A 6, and every additional unit transferred reduces A's retained amount by one. An offer below 4 fails and gives A the outside utility 2 instead. Thus 4 is A's best offer under these preferences and this protocol. There is no need to average A's and B's preferences. B's response first eliminates unsuccessful offers; A then chooses among the remaining consequences for A.
Resolve B's response before comparing A's offers
Outside utilities are 2,4. Among these branches, offering four gives A eight, more than either rejection or the more generous offer.
A full strategy for B says how to respond to every possible offer, including offers that A will not choose. This repeats the distinction between a strategy and a realized path from the sequential lessons. Observing the split 8,4 does not reveal how B would respond to 3 or 10. A prediction obtained by backward reasoning depends on those unobserved responses being optimal under B's specified preferences. The explorer checks the acceptance comparison for the selected offer. It does not elicit an actual person's contingent response strategy or verify that person's beliefs.
Change one assumption on paper and the selected offer can move. If B rejects a tie at 4 but accepts every integer offer above 4, A must offer 5 to obtain agreement and keeps 7. A still prefers this to the outside utility 2. The set of offers weakly better than disagreement has not changed, but the assumed response at equality has. This example explains why a weak inequality in the feasible interval does not itself prove actual acceptance. A behavioral convention was added to select a response at that boundary.
Integer granularity means that offers come in whole units. If arbitrary fractional offers were allowed and B rejected 4 but accepted everything larger, A could offer 4.5, then improve its own retained amount by offering 4.25 instead. Every accepted offer above 4 could be reduced while remaining above 4. There would be no smallest accepted offer under that strict response rule. This is a separate paper game, not a slider setting. The original integer game avoids that particular missing-minimum problem because the next permitted offer after 4 is 5.
Offer-making rights can select opposite endpoints
Protocol
Minimum given to responder
Result A, B
Gains A, B
A proposes once
B gets 4
8, 4
6, 0
B proposes once
A gets 2
2, 10
0, 6
These paper protocols use identical alternatives and acceptance at indifference. Changing the proposer changes selection without changing which divisions are feasible.
Changing who proposes also changes the calculation. In a paper version with B proposing and A accepting at indifference, B offers A its outside amount 2, retains 10, and expects acceptance. Both 8,4 and 2,10 satisfy the original feasibility conditions. They differ because offer-making rights differ, not because the available twelve units changed. This gives a precise meaning to one source of bargaining power. It is the ability to choose terms subject to another party's refusal, under particular response preferences and a particular disagreement outcome.
Compare incentives and outcomes
Locate acceptable splits
AAAAAABBBBBB
A receives 6. B receives 6. Each tile is one unit.
Player
Proposed utility
Outside option
Weakly prefers offer
A
6
2
Yes
B
6
4
Yes
Both weakly accept: Yes. Acceptable integer offers to B: 4, 5, 6, 7, 8, 9, 10.
Bound B’s share
x is the offer to B. a and b are A’s and B’s outside utilities. The inequalities describe feasible agreement, not a unique prediction.
Outside-option presets replace both alternatives but retain the proposed offer. Offer presets and the slider change only B’s share. B accepts at indifference. A’s willingness is reported separately; selecting an offer does not assert it is rational, fair, or actually implemented.
AAAAAAAABBBB
A receives 8. B receives 4. Each tile is one unit.
B receives
A receives
Compared with outside options 2,4
3
9
B prefers disagreement
4
8
Both weakly accept
11
1
A prefers disagreement
The acceptable interval does not select a unique fair allocation.
Use retained offers to separate two acceptance tests
Start the explorer with outside options 2,4 and select Equal split. The twelve tiles show six units each, but the comparison table supplies the reason both weakly prefer agreement. A receives 6 rather than 2; B receives 6 rather than 4. Now use Meet B's outside option. B's share becomes 4 and A's becomes 8. B's acceptance remains Yes, while A's gain increases. These controls change only the proposed allocation. They do not change the alternatives or install a rule that automatically chooses the most profitable accepted offer.
Keep the offer at 4 and select outside options 7,6. The proposed split remains 8,4 because changing alternatives retains the offer. A still prefers 8 to its new outside utility 7, but B now compares 4 with 6 and rejects. Move B's share to 6. B now weakly accepts, yet A receives only 6 and prefers its outside utility 7. The failure has changed sides. This is why separate A and B rows are more informative than one agreement indicator. A proposal can satisfy either party without satisfying both.
Satisfying one party can leave the other unwilling
Outside A, B
Offer B
Split A, B
A willing?
B accepts?
2, 4
4
8, 4
Yes: 8 ≥ 2
Yes: 4 = 4
7, 6
4
8, 4
Yes: 8 ≥ 7
No: 4 < 6
7, 6
6
6, 6
No: 6 < 7
Yes: 6 = 6
The first change retains the offer. The next changes only the offer. Separate comparisons reveal why neither proposal works at alternatives 7,6.
Leave the offer at 6 and switch back to outside options 2,4. Agreement becomes feasible without moving any tiles because the alternatives changed, not the proposed production or division. Now raise B's share to 11. B prefers it, but A retains only 1 and fails its own comparison with 2. Finally set the offer to 10. A retains 2 and is indifferent, while B strictly prefers 10 to 4. The upper endpoint is a limit imposed by A's willingness, just as the lower endpoint comes from B's requirement.
The slider permits these deliberately poor proposals because it is an inspector, not an automatic bargaining agent. A report of B accepts does not mean that A would choose the offer. Nor does an offer with two Yes entries mean that a protocol predicts it. Under the one-shot A-proposer assumptions, 6 is feasible but leaves A less than the feasible offer 4 would. Under a negotiated fairness rule, 6 might instead be selected for an independent reason. The same displayed allocation can therefore support different interpretations only after the selection rule is specified.
Use the zero-option preset to see the role of the lower boundary. Retain an offer of 10 while changing alternatives to 0,0. Both parties now accept, including A with only 2. Set the offer to zero and B still weakly accepts by the stated convention, even though all twelve tiles belong to A. The named offer presets omit duplicate values, so Meet B's outside option need not appear separately when it equals Offer zero. That display choice does not remove zero from the available allocations or alter the tie rule.
Two one-sided requirements overlap at seven offers
Both axes measure the offer to B, not different utility scales. Their common range includes both four and ten under acceptance at indifference.
A useful comparison changes one input while retaining the other. Changing an offer at fixed alternatives tests distribution. Changing alternatives at a fixed offer tests willingness to agree. Changing both together would mix those mechanisms and could conceal which comparison reversed. The explorer's state retention supports that distinction, but it supplies no uncertainty, negotiation history, counteroffer, or automatic optimization. Treat the tables as exact consequences of stipulated preferences. To apply the reasoning elsewhere, first identify the available disagreement outcome and response rule rather than copying a successful slider position.
Agreement is not the same as endorsement
The explorer reports both B's acceptance condition and A's willingness. An intentionally generous or poor offer can be entered even when it is not A's best response. If a + b exceeds 12, no split of this pie gives both their outside utilities. An observed failure to agree may also have other causes, such as private information.
Pareto improvements and fairness judgments are separate. A larger outside option can improve one party's feasible claim without changing physical output. Ask how institutions determine the alternatives, who can credibly walk away, and whether utility is actually linear in the divisible units.
Explain delay and fairness with different assumptions
Feasibility alone cannot explain why some bargains take time. In a paper two-round game, separate from the controls, suppose twelve units are available now, only nine remain tomorrow, both outside options are zero, and B proposes in the final round. Keep whole-unit offers and acceptance at indifference. In the final round B offers A zero and keeps nine. Anticipating that response, B will reject an initial offer below nine. A can therefore offer B nine now and retain three. The present offer is determined by tomorrow's opportunity, not by an equal split.
Compare this with the one-round zero-option protocol, where A offers B zero and keeps twelve. Adding a second opportunity for B changes the current distribution even when agreement occurs immediately and the later round is never reached. The relevant consequence is a credible continuation, not delay that must actually happen. With only six units remaining tomorrow, B's final-round payoff would instead be six, so A could offer six now and retain six. This pair of paper calculations isolates the effect of how valuable waiting is to the responder under the specified final-round rules.
A credible later offer changes today's minimum
Protocol
B gets after rejection
Initial B share
Initial A share
One round, A proposes
0
0
12
Two rounds, 9 remain; B last
9
9
3
Two rounds, 6 remain; B last
6
6
6
Paper protocols only. With zero outside options and tie acceptance, B's final-round opportunity determines the minimum initial offer B will accept.
A delay model needs care about what shrinks. The nine-unit example assumes the available divisible amount physically falls. A different model could retain twelve physical units while each party values tomorrow's receipt less than today's. Those assumptions can have different implications if the parties discount differently. The earlier repetition lesson introduced a future weight, but this explorer has no patience or delay control. Do not read its outside-option numbers as estimated discount factors. They are utilities of disagreement, while a counteroffer continuation is an outcome produced by another strategic interaction.
Preferences can also change acceptance without changing the material units. In a separate paper example, suppose B evaluates an accepted allocation as B's units minus half of A's excess over B when A gets more. With outside utility zero, a 9,3 allocation gives B utility 3 minus half of 6, which is zero. An 8,4 allocation gives B utility 4 minus half of 4, which is 2. A 10,2 allocation gives B utility 2 minus half of 8, which is negative 2. These are stipulated preferences about inequality, not measured human responses.
Under those paper preferences, rejection of 10,2 is consistent with maximizing utility, even though B refuses positive material units. The original explorer would report acceptance at outside option zero because its utility specification is different. Calling the responder irrational would confuse rejection of the analyst's cash-only assumption with inconsistent choice. Conversely, observing rejection does not establish the particular inequality formula just used. A person might have misunderstood the rule or expected future consequences. Distinguishing such explanations requires evidence beyond the allocation and the fact that an offer was rejected.
A material gain can have negative stipulated utility
Material A, B
Half of A excess
B utility calculation
Compare with outside 0
10, 2
4
2 − 4 = −2
Reject
9, 3
3
3 − 3 = 0
Accept by tie rule
8, 4
2
4 − 2 = 2
Accept strictly
This paper preference subtracts half of A's excess from B's receipt. It changes acceptance without claiming that actual responders use this formula.
Pareto efficiency still cannot rank all these divisions. Under the original linear preferences, transferring one unit from A to B helps B and hurts A, so neither adjacent full division Pareto-dominates the other. Both may be efficient within the available allocation set while differing greatly in equality and gains over disagreement. To assess a bargaining institution, state its rule, derive the response incentives, and then evaluate the resulting distribution against an explicit objective. Do not turn a feasible bargain into a forecast, or a forecast into a verdict that the institution is just.
Exercises
Check all acceptable integer splits
The pie is 12 and outside options are A = 5, B = 3. List the acceptable offers to B and the surplus above disagreement.
Hint
B needs at least 3; A must keep at least 5.
Worked answer
AAAAAAAAABBB
A receives 9. B receives 3. Each tile is one unit.
Quantity
Result
B's lower bound
3
B's upper bound
12 − 5 = 7
Total surplus
12 − 5 − 3 = 4
Protocol-selected split A,B
9,3
Separate the interval from the rule selecting a point.
Offers 3,4,5,6,7 are weakly acceptable to both. Their combined outside utility is 8, leaving surplus 4. Under the stated one-shot proposer rule and acceptance at indifference, A would offer 3 and retain 9. That prediction uses the protocol; it is not implied merely by feasibility.
Change a tie rule
Both outside options are zero. Offers are whole units. If B rejects zero but accepts every positive offer, what is A's payoff-maximizing accepted offer?
Hint
Use the smallest allowed strictly positive offer.
Worked answer
AAAAAAAAAAAB
A receives 11. B receives 1. Each tile is one unit.
Offer rule
Smallest accepted offer
A keeps
Integers; zero accepted
0
12
Integers; zero rejected
1
11
Continuous; zero rejected
No minimum positive offer
No maximizing accepted offer by this argument
Action granularity and tie behavior change the solution.
A offers 1 to B and keeps 11. If B instead accepted at indifference, offering zero would leave all 12 with A. With a continuous offer set and rejection at zero, there is no smallest positive offer, so the integer-game conclusion cannot simply be copied.
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 reservation-option, bargaining-power, Pareto, and fairness distinctions. Distributional and ethical claims are not settled by equilibrium arithmetic.