Over the last few years, the vast progress in genome sequencing has highly increased the availability of genomic data. Today, individuals can obtain their digital genomic sequences at reasonable prices from many online service providers. Individuals can store their data on personal devices, reveal it on public online databases, or share it with third parties. Yet, it has been shown that genomic data is very privacy-sensitive and highly correlated between relatives. Therefore, individuals' decisions about how to manage and secure their genomic data are crucial. People of the same family might have very different opinions about (i) how to protect and (ii) whether or not to reveal their genome. We study this tension by using a game-theoretic approach. First, we model the interplay between two purely-selfish family members. We also analyze how the game evolves when relatives behave altruistically. We define closed-form Nash equilibria in different settings. We then extend the game to N players by means of multi-agent influence diagrams that enable us to efficiently compute Nash equilibria. Our results notably demonstrate that altruism does not always lead to a more efficient outcome in genomic-privacy games. They also show that, if the discrepancy between the genome-sharing benefits that players perceive is too high, they will follow opposite sharing strategies, which has a negative impact on the familial utility.