Skip to the main content

Experiment 09 Bargaining

Split It or Walk Away

Divide 100 tokens, test a counteroffer, and see how alternatives, patience, and a deadline change a deal.

5–10 minutes to explore Prototype Updated

Blue and orange playing pieces face a divided tray of counters across a round negotiating table.
About these models Math step by step For experts Math symbol guide

Loading the interactive experiment…

Start with “Work It Out, Step by Step” below. The optional expert section explains its symbols as you go. For more examples, use the plain-language math guide.

1. Make a Prediction

There Are 100 Tokens on the Table

You propose how many of 100 tokens the other side receives. You keep the rest if it accepts. A surplus is extra value created by reaching an agreement. Here the proposed deal has 100 tokens to divide, while each person may also have an outside option: value available from a different plan if this deal fails.

2. Make Your Choice

3. Reveal the Incentives

A Bigger Share of What?

In negotiation mode, the other side accepts when its share is worth at least its outside option and meets its fairness minimum. If it rejects and time remains, it offers you exactly enough tokens to equal your outside option, provided it can still meet its own minimum. That is a deliberately simple response rule, not a solution for every bargaining game.

Patience describes how much a future token is worth compared with a token now. At 90%, each round of waiting multiplies its value by 0.9. A deadline is the last allowed round. When it expires without agreement, both sides take their separate alternatives.

In ultimatum mode, you make one take-it-or-leave-it offer. Rejection gives both sides zero. The fairness minimum still applies, while outside options, patience, and the longer deadline do not. The fairness control represents a chosen acceptance rule, not a claim that one split is objectively fair.

4. Change One Assumption

Change the Alternatives or the Cost of Waiting

Both sides’ settings remain visible. Hiding the threshold is a practice exercise, not private information. Tokens may have different point values for each side. Combined points are shown for comparison within this model, not as a measurement of two people’s happiness. Outside-option points are constant and do not shrink with waiting. Fractional tokens are allowed in the scripted counteroffer. Equal values count as acceptable.

Work It Out, Step by Step

  1. Offer the other side 40 of the 100 tokens. You keep 60.
  2. At the start, a token is worth one point to each side. The deal gives you 60 points and the other side 40.
  3. With a 20-point alternative and a 30-token fairness minimum, the other side accepts 40. Both minimums are met.
  4. After one wait with 90% patience, 40 tokens are worth \(40 \times 0.9=36\) points. After two waits, multiply by 0.9 again: \(36 \times 0.9=32.4\).
  5. If the other side’s alternative is 40 points, 40 tokens after waiting are no longer enough. It needs more tokens to beat or match that alternative.
\[40\times 0.9\times 0.9=32.4.\]

In words: two rounds of waiting multiply the value by nine tenths twice. A seemingly unchanged offer can become less attractive as time passes. An acceptance threshold is the smallest offer that the response rule accepts.

For experts: formal model and assumptions

How to Read the Symbols

\(x\), \(t\), \(a\), \(b\), and \(f\)
\(x\) is tokens offered to the other side. \(t\) counts rounds of waiting, starting at zero. \(a\) and \(b\) are your outside-option points and the other side’s outside-option points. \(f\) is the other side’s fairness minimum in tokens.
\(\delta_A\), \(\delta_B\), and \(u_A,u_B\)
The Greek letter \(\delta\), read “delta,” labels the fraction of value kept per wait. The small A means you; B means the other side. These fractions are called discount factors: 90% patience means a factor of 0.9. \(u\) labels points earned. A small raised \(t\) means multiply the factor by itself \(t\) times; a zero exponent means 1.
\[u_A=(100-x)\delta_A^t,\qquad u_B=x\delta_B^t.\]

In words: multiply each side’s tokens by its remaining fraction of value. A deal is accepted by the scripted recipient when:

\[x\delta_B^t\ge b\quad\text{and}\quad x\ge f.\]

The symbol \(\ge\) reads “greater than or equal to.” These comparisons check the outside option and fairness minimum separately. You can still voluntarily propose a deal worth less than your own alternative; the app points that out. A mutually acceptable split must leave enough value for both alternatives and the recipient’s fairness rule.

The app is a finite, scripted negotiation exercise. It does not calculate an alternating-offer equilibrium or give the simulated player strategic beliefs about future offers. In ultimatum mode, the response depends only on the fairness minimum and rejection pays zero.

See the math reading guide for Greek letters, exponents, and comparisons.

5. Transfer the Lesson

A Deadline Is Part of the Deal

Two teams are negotiating how to divide a project budget. One has another funded project available; the other faces an approaching deadline. Which parts of this model might explain their choices? Which relationships or fairness concerns are missing from the point totals?

Concept reference: Yale’s lecture on ultimatums and bargaining. The acceptance and counteroffer scripts here are specific teaching assumptions.

A Model Is a Place to Start

These small models make the incentives visible. Their results follow from their stated rules; they are not forecasts of how every person or organization behaves. A simulated strategy is a rule, not a personality.

Scenario links save the controls and random seed. To reproduce an interactive run, make the same choices in the same order. Changing a setting restarts the experiment.