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Game theory can provide valuable insights into management of multi-player systems. Players (agents), who seek maximizing their utilities, change their preferences and strategies in response to the social, political, economic and environmental changes over time. Overlooking the dynamic structure of the games can result in misleading policy insights and development of highly undesirable outcomes. Using a simple natural resource sharing game, this paper shows how rational players might change their behavior in response to the changing conditions of the game. It is argued that recognition of the possible evolution path of the games can help justify certain decisions by the players in various stages of the game and develop timely measures to prevent undesirable loss-loss situations.