Question
Two siblings, Ada and Ben, are dividing the $100 proceeds of a market stall. The money is locked into ten $10 vouchers, and every proposed split must allocate whole vouchers; neither sibling can make side payments.
Ada proposes a split now (round 0), Ben may propose one in round 1 if the first is rejected or Ada passes, and Ada has a final turn in round 2. Rejection or passing advances one round. If the round-2 offer fails, an automatic settlement gives each sibling five vouchers in round 3. A successful proposal is paid immediately in its round. Ada does not discount later receipts; Ben values each voucher received one round later at 80% of its present value. Both maximize their own discounted receipts, responders accept if indifferent, and a proposer who is indifferent between securing an acceptable offer and passing chooses to pass.
Work backward to find the best immediately acceptable opening split and what actually happens. Would the outcome change if the $100 could instead be split into individual dollars?
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Hint 1
Start at the last proposal. Ben compares a round-2 offer with the value, one round earlier, of his five-voucher fallback. Round up any required transfer to whole vouchers.
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Solution
All values below are measured in vouchers at the round in question. Only Ben discounts, by \(\delta_B=0.8\); Ada's factor is \(\delta_A=1\).
Round 2 (Ada proposes). If Ben rejects, his round-3 fallback is five vouchers, worth \(0.8(5)=4\) in round 2. The least acceptable whole-voucher offer gives Ben four and Ada six. Ben accepts on equality. Ada prefers six now to her fallback of five, so this offer is made.
Round 1 (Ben proposes). If Ada rejects, she receives six vouchers in round 2 and, with no discounting, values that at six in round 1. Ben must therefore give Ada at least six. He keeps four, strictly better than waiting to receive four in round 2, which is worth \(0.8(4)=3.2\) in round 1. Thus Ben offers \((\text{Ada},\text{Ben})=(6,4)\), accepted by Ada on indifference.
Round 0 (Ada proposes). Ben can get four in round 1, worth \(0.8(4)=3.2\) now. Because transfers are indivisible, the smallest acceptable current transfer is \(\lceil3.2\rceil=4\) vouchers. That leaves Ada six. Passing also gives Ada six in round 1, with no discount, so she is indifferent. Under the specified proposer tie rule she passes, and Ben's round-1 offer of \((6,4)\) is accepted. An accepted opening offer was possible; delay is selected by the tie rule, not forced by rejection.
With dollar-level rather than $10-voucher indivisibility, the round-2 split is $60/$40, and the round-1 split is again $60/$40. At round 0, Ben needs only \(0.8(\$40)=\$32\); Ada can keep $68, which strictly beats the $60 she gets from passing. She offers $68/$32 and Ben accepts. The coarse transfer unit removes Ada's strict gain from an immediate agreement. In general, for a continuation split of \(a+b=N\) indivisible units, the minimum acceptable current offer to a responder discounted by \(\delta_B\) is \(\lceil\delta_B b\rceil\); compare the proposer's \(N-\lceil\delta_B b\rceil\) with their own discounted continuation \(\delta_A a\). When \(\delta_A=1\) and \(\lceil\delta_B b\rceil=b\), those payoffs tie, permitting delay under the stated tie rule.
Assumptions
Exactly ten vouchers must be allocated on every accepted proposal.
Only Ben discounts: a voucher paid one round later is worth 0.8 voucher to him.
Fallback occurs in round 3 if no proposal is accepted; acceptance on equality and proposer passing on indifference are binding tie rules.
For the follow-up, fractional $10 vouchers are not permitted in the base game; whole-dollar splits are permitted instead.
Final Answer
With $10 vouchers, Ada's best acceptable round-0 split is 6 vouchers for Ada and 4 for Ben, but Ada is indifferent to passing and therefore passes. In round 1, Ben offers Ada 6 and Ben 4; Ada accepts. With $1 divisibility, Ada instead offers $68 to herself and $32 to Ben in round 0, and Ben accepts immediately.