Convexity of the triple helix of innovation game

Abstract

Purpose This paper aims to determine the conditions for the core of the Triple Helix game to exist. The Triple Helix of university-industry-government relationships is a three-person cooperative game with transferable utility. Then, the core, the Shapley value and the nucleolus were used as indicators of the synergy within an innovation system. Whereas the Shapley value and the nucleolus always exist, the core may not. Design/methodology/approach The core of a three-person cooperative game with transferable utility exists only if and only if the game is convex. The paper applies the convexity condition to the Triple Helix game. Findings The Triple Helix game is convex if and only if there is output within the system; it is strictly convex if and only if all the three bilateral and the trilateral relationships have an output. Practical implications Convex games are competitive situations in which there are strong incentives towards the formation of large coalitions; therefore, innovation actors must cooperate to maximise their interests. Furthermore, a Triple Helix game may be split into subgames for comprehensive analyses and several Triple Helix games may be combined for a global study. Originality/value This paper extends the meaning of the Shapley value and the nucleolus for Triple Helix innovation actors: the Shapley value indicates the quantity a player wins because of the coalitions he involves in and the nucleolus the return for solidarity of an innovation actor.

The Triple Helix of innovation as a double game involving domestic and foreign actors

“…Thirdly, Shapley (1965Shapley ( , 1971) stated that a convex game is decomposable (therefore, may split into its finest components) and demonstrated that a strictly convex game is indecomposable (see also González-Díaz & Sánchez-Rodríguez, 2008). Because a Triple Helix game where all the bilateral and the trilateral relationships exist is strictly convex (cf., Mêgnigbêto, 2024), it is indecomposable. In conclusion, neither the global Triple Helix game is the summation or the combination of the domestic and foreign Triple Helix games, nor the Triple Helix domestic and foreign games are components of the global Triple Helix game.…”

“…Let us individually consider each layer (its actors and their collaboration relations at the level): it is an entire Triple Helix relationship, so a game (Mêgnigbêto, 2018c); hence, the Triple Helix game at an area level may be analyzed as bringing together two games, the domestic and the foreign one, with the same actors (university, industry and government) that may play either at the domestic or at the foreign level. Mêgnigbêto (2024) showed that a Triple Helix of innovators game is convex; therefore, the global, domestic, and foreign (above-mentioned) games all are convex.…”

“…The core of a Triple Helix game exists, provided that there is output within the system (Mêgnigbêto, 2024); when all the bilateral and the trilateral relations exist within the Triple Helix innovation system, the game is strictly convex and the core is a hexagon, hence, its surface area could be used as a measure of the synergy within a Triple Helix innovation system (Mêgnigbêto, 2024).…”

“…The games studied are all strictly convex; thus, each core has the form of a hexagon (Mêgnigbêto, 2024). The width is mainly determined by the contribution of the actor i and the length by the contributions of both u and g (Table 4) (Mêgnigbêto, 2018c(Mêgnigbêto, , 2018b(Mêgnigbêto, , 2019.…”