Non-Cooperative Games
1951 · paper · Mathematics and statistics
Introduces non-cooperative games, in which each participant acts independently without coalitions, and defines equilibrium points. Proves that every finite non-cooperative game has at least one equilibrium point, studies the structure of solutions, and works an example of a simplified three-person poker game.
Checked: PDF from library.princeton.edu, fetched 4 Oct 2026. Page numbers are those of that PDF.
2 findings
Substance Proof of Theo. 3 (printed p.11): the second inequality reads "P_j(t; S_i) >= P_j(t; S_i'; r_j)"; its left side should be P_j(t; S_i') (prime missing), otherwise it does not follow from criterion (1) and the two inequalities do not add to the stated P_j(t; S_i*) >= P_j(t; S_i*; r_j).
- p. 15 Page 11, proof of Theo. 3, displayed pair of inequalities
P_j(t; S_i) ≥ P_j(t; S_i'; r_j)
Judge's note: Viewed PDF p.15 and a 4x crop of the displayed line rendered from the PDF. The first inequality is P_j(t;S_i) >= P_j(t;S_i;r_j). In the second, the right side has a clear prime on S_i, and the left side has none (clean background, no faint mark). With S_i* = (S_i+S_i')/2 the sum needs P_j(t;S_i) + P_j(t;S_i') on the left, so the left side of the second inequality must be primed. The handwritten formula omits the prime. The proof's conclusion is unaffected.
Second check: another model agreed · checked by hand
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Substance Printed p.19: "V3 = .243 = 79/40 ( (1-alpha)/(5+alpha) )" - with alpha = (21 - sqrt(321))/10 = .308 the closed form evaluates to about .257, not .243; the decimal .243 equals -(V1+V2) and is given by (79 - 89 alpha)/(40(5+alpha)).
- p. 24 Page 19, values of the poker game, v3
V3 = .243 = 79/40 ( (1-α)/(5+α) )
- p. 23 Page 18, solution for α
α = (21 - √321)/10
- p. 24 Page 19, v1 and v2
V1 = -.147 = -(1+17α)/8(5+α) , V2 = -.096 = -(1-2α)/4
- p. 23 solution for α
α=.308
Judge's note: Read the formulas on an enlarged crop of PDF p.24 (79/40 times (1-alpha)/(5+alpha)) and alpha on PDF p.23. With sqrt(321) = 17.916473, alpha = 0.30835. The printed closed forms for V1 (denominator 8(5+alpha)) and V2 reproduce -.147 and -.096, so alpha is right. The V3 closed form gives 0.2573. The game is zero-sum (rules 2, 4, 5 on PDF p.22: chips are anted and the pot is divided among the players), and -.147 - .096 + .243 = 0, so the decimal is right and the closed form is wrong. -(V1+V2) simplifies, using 5 alpha^2 - 21 alpha + 6 = 0, to (79 - 89 alpha)/(40(5+alpha)) = 0.2428. No statement elsewhere explains the gap; it is far beyond rounding. The replacement formula is a derivation, not something printed in the paper.
Second check: another model agreed
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12 minor (typos and the like; most checked by the judge only)
Reference or label The page numbered "-12a-" ("Simple Examples") sits between printed p.13 (ending "Therefore, the") and p.14 (beginning "complete set [which is finite] of conditions"), splitting the proof of Theo. 5; the "Simple Examples" section has no entry in the Table of Contents.
- p. 18 Page header
-12a-
- p. 17 Page 13, last line
Therefore, the
- p. 19 Page 14, first line
complete set [which is finite] of conditions
- p. 4 Table of Contents
6. Geometrical Form of Solutions
Judge's note: Viewed PDF pp.16-19 and the contents page. PDF p.16 is -12-, p.17 is the start of Geometrical Form of Solutions and ends mid-sentence "Therefore, the", p.18 is -12a- Simple Examples, p.19 is -14- and continues the sentence. So 12a is out of place in this PDF; it belongs after p.12. The contents list has twelve entries and none for Simple Examples. This may be a collation fault of the scanned copy rather than the author's. The third reviewer's sub-claim is not upheld: the contents entry for Section 6 reads 13 on an enlarged crop, and the faint handwritten number on PDF p.17 is consistent with 13, so that pointer is not shown to be wrong.
Second check: another model agreed
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Wording Proof of Theo. 1 (printed p.6): the fixed-point equation is written with the n-tuple symbol on the left, "s_mu = Sum_alpha pi_i,alpha C'_i,alpha(s_mu, lambda(mu))", while the right side is only player i's mixed strategy (the i-th component, cf. the definition of S_i'(s,lambda) on printed p.5).
- p. 10 Page 6, proof of Theo. 1, displayed equation before "because s_μ is a fixed point"
s_μ = Σ π_iα C'_iα(s_μ, λ(μ))
Judge's note: Viewed PDF p.10 and the definition on PDF p.9. The left-hand symbol is the same script letter used for the n-tuple in the same line ("because s_mu is a fixed point"), and the right side is the sum that p.9 defines as S_i'. So the equation strictly holds for the i-th component of s_mu. The meaning is clear and the argument is unaffected; this is loose notation rather than a wrong result.
Second check: another model agreed
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Typo Misspelling: "Important definitions will be preceeded by a sub-title indicating the concept defined" should read "preceded".
- p. 6 Page 2, first paragraph
preceeded by a sub-title
- p. 6 Formal Definitions and Terminology, first paragraph
preceeded by a sub-title indicating the concept defined
Judge's note: Read on the page image of PDF p.6 (first paragraph of Formal Definitions and Terminology): the typescript clearly shows "preceeded".
Second check: none (judge only)
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Typo Misspelling: "pieces of albegraic varieties, cut out by other algebraic varieties" - "albegraic" should be "algebraic".
- p. 8 Page 4, last paragraph
pieces of albegraic varieties
- p. 8 printed p. 4, last paragraph
pieces of albegraic varieties, cut out by
Judge's note: Read on the page image of PDF p.8 (printed p.4, last paragraph). "albegraic" is typed with b and g transposed; the next line has the correct "algebraic".
Second check: none (judge only)
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Typo Theorems are numbered out of order of appearance: "THEO. 1" (printed p.5), "THEO. 4: Any finite game has a symmetric equilibrium point." (p.8), "THEO. 2" and "THEO. 3" (p.10), "THEO. 5" (p.13).
- p. 12 Page 8, Symmetries of Games
THEO. 4: Any finite game has a symmetric equilibrium point.
- p. 14 Page 10, Solutions
THEO. 2: A sub-solution
- p. 9 Page 5, Existence of Equilibrium Points
THEO. 1: Every finite game has an equilibrium point.
- p. 17 Geometrical Form of Solutions, printed p. 13
THEO. 5: The sets
Judge's note: Checked the theorem labels on the images of PDF pp.9, 12, 14 and 17. The order is 1, 4, 2, 3, 5. The only cross-reference found ("from Theo. 2", PDF p.17) points to the right theorem, so nothing is broken; the numbering is simply not sequential.
Second check: none (judge only)
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Typo Misspelling "paramenters" for "parameters": "of the greek paramenters must vanish" (printed p.18) and "Paramenters not mentioned are zero" (printed p.19).
- p. 23 Poker section, printed p. 18, paragraph below the table
of the greek paramenters must vanish
- p. 24 Poker section, printed p. 19, paragraph before the coalition tables
Paramenters not men-
Judge's note: Enlarged crops of both lines. PDF p.23 clearly reads "paramenters". The PDF p.24 instance is in a degraded part of the scan; it has the same word length and shape ("Paramenters") but is less certain. The previous page (PDF p.22) spells "behavior parameters" correctly.
Second check: none (judge only)
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Typo Subject-verb disagreement: "The basic requirements for a non-cooperative game is that there should be no pre-play communication among the players" ("requirement ... is" intended; only one requirement follows).
- p. 26 Page 21, second paragraph
The basic requirements for a non-cooperative game is that
- p. 26 Motivation and Interpretation, second paragraph
The basic requirements for a non-cooperative game is that there
Judge's note: Read on the page image of PDF p.26 (printed p.21, second paragraph): "requirements" and "is" are both clearly typed.
Second check: none (judge only)
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Typo Misspelling: "Thus the assumptions we made in this "mass-action" interpretion lead to the conclusion" - "interpretion" should be "interpretation".
- p. 27 Page 22, paragraph beginning "Thus the assumptions"
"mass-action" interpretion
Judge's note: Read on the page image of PDF p.27 (printed p.22). The word is typed "interpretion"; PDF p.26 has the correct "interpretation" twice.
Second check: none (judge only)
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Typo Misspelling: "we can only expect some sort of approximate euqilibrium" - "euqilibrium" should be "equilibrium".
- p. 28 Page 23, second paragraph
approximate euqilibrium
Judge's note: Enlarged crop of PDF p.28 (printed p.23, second paragraph): the word begins "euq", with the u before the q.
Second check: none (judge only)
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Typo Printed p. 23 (PDF p. 28): "prediction of the behavior to be expected of rational playing the game in question?" lacks a noun after "rational" (e.g. "rational men" / "rational players").
- p. 28 Page 23, paragraph beginning "We proceed by investigating"
expected of rational playing the game in question
Judge's note: Read the full page image and an enlarged crop of the line. The typescript reads "to be expected of rational playing the game in" within a single line, so nothing is lost at a line break, and there is no handwritten insertion. The next paragraph on the same page has "rational men playing in position i", which shows the intended form. A missing word, cosmetic.
Second check: none (judge only)
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Typo Printed p. 23 (PDF p. 28): "euqilibrium, since the information, its utilization" — "equilibrium" is typed with q and u transposed.
- p. 28 Motivation and Interpretation, printed p. 23, second paragraph
euqilibrium, since the information, its utilization
Judge's note: Read the full page image and an enlarged crop of the line. The word at the start of the line is clearly typed "euqilibrium" (e-u-q-i), while the same page spells "equilibrium" correctly further down ("sets of equilibrium strategies", "set of equilibrium points"). The transposition is in the typescript, not an OCR artefact. Cosmetic misspelling.
Second check: none (judge only)
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Typo Printed p. 26 (PDF p. 31): "Thus the problem analyzing a cooperative game becomes the problem of obtaining" — "of" is missing after the first "problem".
- p. 31 Page 26, first paragraph
Thus the problem analyzing a cooperative game becomes
- p. 31 Applications, printed p. 26, first paragraph
Thus the problem analyzing a cooperative game becomes the problem of obtaining
Judge's note: Read the full page image. The line reads "Thus the problem analyzing a" and continues "cooperative game becomes the problem of obtaining a suitable, and convincing, non-cooperative model". No "of" is typed and there is no handwritten insertion on this line (the only hand marks on the page are the brackets around "extended to infinite games"). The parallel phrase "the problem of obtaining" in the same sentence shows the intended construction. A missing word, cosmetic.
Second check: none (judge only)