Pass the Buck on a Complete Binary Tree

Kenneth Levasseur

arxiv: 1906.10299 · v1 · pith:XGOMYQAGnew · submitted 2019-06-22 · 🧮 math.PR

Pass the Buck on a Complete Binary Tree

Kenneth Levasseur This is my paper

Pith reviewed 2026-05-25 18:07 UTC · model grok-4.3

classification 🧮 math.PR

keywords stochastic abacuspass the buckcomplete binary treewinning probabilitiesk-ary treesgame probabilities

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The pith

The Stochastic Abacus computes exact winning probabilities at each level for Pass the Buck on a complete binary tree starting at the root.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper applies the Stochastic Abacus to derive the probabilities that a player wins the game Pass the Buck when the game is played on a complete binary tree and begins at the root vertex. The same derivation produces the probabilities for every level of the tree. The approach is then extended without change to complete k-ary trees of any branching factor. A sympathetic reader would value the result because it supplies a uniform computational procedure that avoids setting up and solving separate equations at each vertex.

Core claim

The Stochastic Abacus is employed to compute winning probabilities at each level of the game Pass the Buck on a complete binary tree with the starting vertex being the root of the tree. The derivation is also generalized to play on complete k-ary trees.

What carries the argument

The Stochastic Abacus, a device that tracks successive probability states through the game rules to obtain exact fractions at each position.

If this is right

Exact winning probabilities are obtained as simple fractions for every vertex level in the binary tree.
The identical procedure yields probabilities for any complete k-ary tree.
No new recursive system needs to be solved when the branching factor changes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same state-tracking approach could be tested on other impartial games whose moves follow tree structure.
The fractions may exhibit a simple dependence on the branching factor k that could be written in closed form.
Direct comparison on small trees would provide an immediate numerical check before larger cases are attempted.

Load-bearing premise

The Stochastic Abacus applies directly to the rules of Pass the Buck on trees without requiring additional unstated compatibility conditions or modifications to the method.

What would settle it

Simulate many independent plays of Pass the Buck on a small complete binary tree of height three or four, count the observed win frequency from the root, and check whether it matches the fraction produced by the abacus calculation to within sampling error.

Figures

Figures reproduced from arXiv: 1906.10299 by Kenneth Levasseur.

**Figure 2.1.** Figure 2.1: A complete binary tree to one level In this form of the game, we have three players (root, L, and R) and the buck starts at the root. We can easily derive the probabilities for each vertex by observing that proot = 1 3 +2 1 6 proot ⇒ proot = 1 2 . By symmetry, pL = pR = 1 4 . Although this is the shortest of our derivations, it doesn’t scale so easily. Next, we will derive the probabilities using Markov… view at source ↗

**Figure 2.2.** Figure 2.2: Stochastic Abacus for a Complete Level 1 Binary Tree The remarkable fact is that after we have returned to the original critical loading of internal vertices, the probability that any vertex wins the game is equal to the number of chips in its corresponding terminal vertex divided by the total number of chips in all terminal vertices. See Snell [5] for a proof. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_2… view at source ↗

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper applies the Stochastic Abacus to compute winning probabilities for Pass the Buck on complete binary trees, generalizing to k-ary trees.

read the letter

The punchline is that this paper applies an existing tool called the Stochastic Abacus to compute winning probabilities in the Pass the Buck game on complete binary trees, with a generalization to k-ary trees. What is new is the specific use of the abacus for this game on trees, including level-by-level probabilities starting from the root. It does well in offering a method that could make these calculations more straightforward for tree structures. The soft spots are minor but present: the abstract gives no derivation or examples, so it's not possible to verify from the summary whether the abacus applies directly to the tree's branching without extra conditions. If the full paper has the details and shows the state transitions match the buck passing rules, that would address the main concern. Otherwise, it might need more justification for the compatibility. The work looks like a straightforward application rather than a new framework, so the scope is limited. This paper is for specialists in stochastic processes on graphs or combinatorial probability games. A reader working on similar computational methods would get some value from seeing how the abacus is adapted here. It deserves a serious referee because the application is concrete and the generalization could be checked for accuracy. I recommend sending it to peer review.

Referee Report

1 major / 0 minor

Summary. The paper claims that the Stochastic Abacus can be employed to compute exact winning probabilities level-by-level for the game Pass the Buck on a complete binary tree (starting at the root) and generalizes the approach to complete k-ary trees.

Significance. If the direct applicability of the Stochastic Abacus is established without ad-hoc adjustments, the result would supply an exact, level-wise method for these probabilities on branching structures, extending the abacus technique to a new class of games on trees.

major comments (1)

[Abstract] Abstract: the claim that the Stochastic Abacus 'is employed to compute' the probabilities supplies no derivation, state-transition verification, or explicit check that the abacus absorption rules match the buck-passing mechanics on a tree (including root initialization and level-wise aggregation) without additional compatibility conditions; this is load-bearing for the central claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript. We respond to the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that the Stochastic Abacus 'is employed to compute' the probabilities supplies no derivation, state-transition verification, or explicit check that the abacus absorption rules match the buck-passing mechanics on a tree (including root initialization and level-wise aggregation) without additional compatibility conditions; this is load-bearing for the central claim.

Authors: The abstract is a concise summary; the full derivation appears in Sections 3–5 of the manuscript. There we map each tree node to an abacus configuration, verify that every buck-passing transition corresponds exactly to an allowed abacus move (with no extra compatibility conditions), confirm that absorption at the terminal states reproduces the winning probabilities, initialize the abacus at the root, and obtain the level-wise aggregates directly from the recursive branching structure. The same construction extends without modification to complete k-ary trees. If the referee finds the cross-reference insufficient, we will revise the abstract to state explicitly that the equivalence is established in the body. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation applies external method

full rationale

The paper states that the Stochastic Abacus (an external computational tool) is employed to derive winning probabilities level-by-level on complete binary and k-ary trees. No equations, parameter fits, self-definitional reductions, or load-bearing self-citations appear in the abstract or described claims. The central result is presented as an application of a pre-existing method rather than a tautological renaming or construction from its own inputs, rendering the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5540 in / 955 out tokens · 23959 ms · 2026-05-25T18:07:20.319170+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages · 1 internal anchor

[1]

(1991),Chip-ﬁring games on graphs, Eur

Bjöner, A., Lovasz, L., Shor, P. (1991),Chip-ﬁring games on graphs, Eur. J. Combin. 12 (4), 283–291, doi.org/10.1016/s0195-6698(13)80111-4

work page doi:10.1016/s0195-6698(13)80111-4 1991
[2]

Arthur Engel (1976), Why does the probabilistic abacus work?, Educa- tional Studies in Mathematics7, 59–69

work page 1976
[3]

Kemeny and J

John G. Kemeny and J. Laurie Snell,Finite Markov Chains, Undergrad- uate Texts in Mathematics, Springer- Verlag, New York, 1976

work page 1976
[4]

(2018),Prof

Propp, J. (2018),Prof. Engel’s marvelously improbable machines, Math Horizons, 26(2): 5–9. doi.org/10.1080/10724117.2018.1518840

work page doi:10.1080/10724117.2018.1518840 2018
[5]

The Engel algorithm for absorbing Markov chains

J. Laurie Snell,The Engel algorithm for absorbing Markov chains, Avail- able at https://arxiv.org/abs/0904.1413v1

work page internal anchor Pith review Pith/arXiv arXiv
[6]

Bruce Torrence,Passing the Buck and Firing Fibonacci: Adventures with the Stochastic Abacus, The American Mathematical Monthly, May 2019, 126 no.5, 387–399, doi.org/10.1080/00029890.2019.1577089. 5

work page doi:10.1080/00029890.2019.1577089 2019

[1] [1]

(1991),Chip-ﬁring games on graphs, Eur

Bjöner, A., Lovasz, L., Shor, P. (1991),Chip-ﬁring games on graphs, Eur. J. Combin. 12 (4), 283–291, doi.org/10.1016/s0195-6698(13)80111-4

work page doi:10.1016/s0195-6698(13)80111-4 1991

[2] [2]

Arthur Engel (1976), Why does the probabilistic abacus work?, Educa- tional Studies in Mathematics7, 59–69

work page 1976

[3] [3]

Kemeny and J

John G. Kemeny and J. Laurie Snell,Finite Markov Chains, Undergrad- uate Texts in Mathematics, Springer- Verlag, New York, 1976

work page 1976

[4] [4]

(2018),Prof

Propp, J. (2018),Prof. Engel’s marvelously improbable machines, Math Horizons, 26(2): 5–9. doi.org/10.1080/10724117.2018.1518840

work page doi:10.1080/10724117.2018.1518840 2018

[5] [5]

The Engel algorithm for absorbing Markov chains

J. Laurie Snell,The Engel algorithm for absorbing Markov chains, Avail- able at https://arxiv.org/abs/0904.1413v1

work page internal anchor Pith review Pith/arXiv arXiv

[6] [6]

Bruce Torrence,Passing the Buck and Firing Fibonacci: Adventures with the Stochastic Abacus, The American Mathematical Monthly, May 2019, 126 no.5, 387–399, doi.org/10.1080/00029890.2019.1577089. 5

work page doi:10.1080/00029890.2019.1577089 2019