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We maximize the <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\beta $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03b2<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-optimistic value of each generator\u2019s uncertain profit function in this game. Emphasis is first put on computing the <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\beta $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03b2<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-optimistic value of the uncertain profit function through the inverse uncertainty distribution of the uncertain variable in the inverse demand function. We later focus on transforming the single-period spatial oligopolistic game, which is a generalized Nash equilibrium (short for GNE ) problem, into a variational inequality problem. Then the model is applied to simulate the transmission price of the electricity system of America, which reveals that our model is effective.<\/jats:p>","DOI":"10.1007\/s00500-019-04665-1","type":"journal-article","created":{"date-parts":[[2020,1,13]],"date-time":"2020-01-13T14:02:44Z","timestamp":1578924164000},"page":"2887-2895","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A spatial oligopolistic electricity model under uncertain demands"],"prefix":"10.1007","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5984-3323","authenticated-orcid":false,"given":"Qiqiong","family":"Chen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuanguo","family":"Zhu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,1,13]]},"reference":[{"issue":"2","key":"4665_CR1","doi-asserted-by":"publisher","first-page":"94","DOI":"10.1016\/0140-9883(92)90002-U","volume":"14","author":"F Bolle","year":"1992","unstructured":"Bolle F (1992) Supply function equilibria and the danger of tacit collusion: the case of spot markets for electricity. 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