According to philosophical presumptions of game theory, are people trustworthy if there are incentives (profit and less loss) involved in lying? And if this question isn't entirely accurate as a philosophical question, where are the faulty assumptions?
People have incentives to tell the truth, and to lie. Agents within a system will use different strategies, and face different consequences. The whole point of game theory is different Nash equilibria might emerge, depending what strategies predominate.
If total trust is assumed, but violation of trust has few consequences, that incentivises lying, and could allow catastrophic consequences from one liar. On average, higher levels of trust will allow for more cooperation, and we would expect cultural systems that help ensure that to spread - eg the 'panopticon' quality of believing we will be judged at death will spread (& is close to a cultural universal).
Trustworthiness is a strategy. Agents will have a mix of strategies.
Game theory models how people behave in terms of incentives. That's useful in a wide variety of contexts, whether talking about ecological systems, individuals, corporations, or nations. "Game theory" doesn't have any opinion on whether people are trustworthy anymore than it has an opinion on what color cars should be.
However, if you use game theory for something, then you are taking the stance that it's useful to model or predict people's behavior in terms of their incentives. If their incentives don't include anything that you would consider "trustworthiness", then you probably don't consider them that trustworthy. Of course, you could also model trustworthiness as an incentive.
From here, you can ask more philosophical questions. If you define a system in which cheating is punished, and people behave, does that count as trustworthy? Maybe as the rules designer, you need to earn the people's trust for them to follow the rules. Tough questions!
I'll give you a short, opinionated answer: people are more selfish than trustworthy but they are capable of coordinating broadly according to their relationships, laws, culture and religious beliefs, which makes them somewhat trustworthy.
First, this is not a question of human nature and people, but a model of modeling human behavior with the mathematical models of game theory. So while it may seem pedantic, it is important to distinguish between actual human people and the fictional agents that model people. However, like real life which it attempts to model, generally game-theoretic agents (fictional people) are not trustworthy in the face of a preponderance of incentives to lie (like actual people are known to do).
According to the WP article on game theory:
Game theory is the study of mathematical models of strategic interaction among rational decision-makers.
Again, there's an important philosophical distinction to be made between models and reality. It is a plain fact that the map is not the territory. The question, therefore, is 'Is it rational for a decision-maker to lie if there are adequate incentives?' The answer to this is a simple 'yes' with the caveat that one properly defines what it means to be a rational decision-maker. This is yet another important philsophical distinction, because it hinges upon the philosophical question 'What is reason?' This is no small ask since human reason is defeasible and somewhat difficut to characterize. The most widely used term I am familiar with to model an actual person with a fictional model in this context of modeling human behavior is the pseudo-Latin *Homo economicus*. From the article:
The term homo economicus, or economic man, is the portrayal of humans as agents who are consistently rational, narrowly self-interested, and who pursue their subjectively-defined ends optimally. It is a word play on Homo sapiens, used in some economic theories and in pedagogy.3... The rationality implied in homo economicus does not restrict what sort of preferences are admissible. Only naïve applications of the homo economicus model assume that agents know what is best for their long-term physical and mental health. For example, an agent's utility function could be linked to the perceived utility of other agents (such as one's husband or children), making homo economicus compatible with other models such as homo reciprocans, which emphasizes human cooperation.
Thus, the answer to your question is generalized because if one addresses the nature of preference, one can model altruism and have a rational decision-maker embody Homo reciprocans. In fact, there are entire theories devoted to forms of altruism like the biological altruism of sociobiology. The fact that human beings are not actually well-modeled by Homo economicus is a major thesis of Daniel Kahneman and Amos Tversky and is discussed at length in Thinking, Fast and Slow.