phenomenon · claim-type · cross-domain-transfer · mechanism · claim-relation · replication-status · limitation — grounded in the full text| phenomenon | equilibrium points in n-person games— “A self-countering n-tuple is called an equilibrium point.” | |
| claim-type | existence— “Hence there is an equilibrium point.” | |
| method — evidence | — “Since the graph is closed and since the-image of each point under the mapping is convex, we infer from Kakutani's theorem' that the mapping has a fixed point (i.e., point contained in its… | |
| transfer | from topology / fixed-point theory (Kakutani): The existence proof imports Kakutani's fixed-point theorem for convex-valued closed-graph set correspondences and applies it to the game's countering correspondence. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | transfers[0] |
| mechanism | Because the countering correspondence is convex-valued and has a closed graph, a fixed-point theorem applies to it, and any fixed point of that correspondence is by definition a self-countering n-tuple, i.e. an equilibrium. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | mechanisms[0] |
| builds-on | Kakutani's fixed point theorem Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | claim_relations[0] |
| replication | untested | |
| limitation | Equilibrium points are not interchangeable in general: outside the two-person zero-sum case, distinct equilibrium points need not give the players the same expectations. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[0] |
finding · quantity · variable · definition · key-term — grounded in the full text| finding | Every finite n-person game has at least one equilibrium point. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[0] |
| finding | The set of n-tuples that counter a given n-tuple is convex. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[1] |
| finding | Continuity of the pay-off functions implies that the graph of the countering correspondence is closed. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[2] |
| finding | Outside the two-person zero-sum case, different equilibrium points need not give the players the same expectations. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[3] |
| evidence status note | finding — THE QUOTE'S OPENING WAS MOVED BY THE TEXT EXTRACTION: the paper's sentence begins 'In this case any two equilibrium' (printed earlier in this copy, inside the sentence before), and 'this case' is the two-person zero-sum case. | content_tags.findings[3] |
| finding | In the two-person zero-sum case, existence of an equilibrium point is equivalent to von Neumann's main theorem. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[4] |
| definition | n-person game Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[0] |
| definition | equilibrium point Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[1] |
| definition | pay-off functions Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[2] |
| definition | mapping of the product space Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[3] |
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