In 1944 John von Neumann and Oskar Morgenstern published Theory of Games and Economic Behaviour which was a breakthrough in the area of Mathematics and Economics. It fostered thinking about games in a more rational way, and gave us a powerful framework in which to do so. Having read it, I would like to share some of my thoughts and perhaps show you how you can apply these concepts to startups or, indeed, to everyday life. The techniques at the heart of this theory, or maths in general, can give us some basic intuitions for thinking about startup strategies, since Game Theory investigates how to behave when there is a conflict of interest.
Prisoner's Dilemma
Testify or refuse? You may have heard of the famous Prisoner’s Dilemma  an example of a nonzerosum game where one player winning does not mean that a second player loses. I would like to point out some conclusions that this game might lead us to. Let’s have a look at the scenario:
Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge. They hope to get both sentenced to a year in prison on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to: betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent. The offer is:

If A and B each betray the other, each of them serves 2 years in prison

If A betrays B but B remains silent, A will be set free and B will serve 3 years in prison1 (and vice versa)

If A and B both remain silent, both of them will only serve 1 year in prison (on the lesser charge)
They can choose to keep quiet or to talk with the prosecutors. The prisoners can make assumptions in order to maximize their returns and spend less time in a prison if the other thinks in the same way. The best result for both prisoners is to cooperate with the prosecutors and receive a lower sentence — a better solution compared with the case when one of them chooses to testify and the second refuses. When both of them testify, this is referred to as the Nash equilibrium. Betrayal is the dominant strategy because it is the best option for the prisoners regardless of which the other chooses.
This conflict illustrates how to pick a solution and how to think about strategy. It gives us some insight which can be useful when thinking about situations we may face in business or in life. The Prisoner’s Dilemma allows you to model events, for instance when an employee demands a raise or when somebody wants to buy car and there is a risk that it could be faulty.
Nash equilibria
John Nash was an American mathematician and economist who studied Game Theory. He was creative and, despite the difficulties he faced through mental illness, formulated a theory which opened up a new chapter in Mathematics, ultimately winning him the Nobel Prize in Economy. His story was told in the movie A Beautiful Mind:
Nash theorised about equilibria based on noncooperative games in which players seek strategies to maximise their winnings given that their opponents will also use strategies designed to maximise their winnings.
Intuitions
A strategy is set of sequences of actions that specify the player’s choices at ALL possible decision points in a game and incorporates information about the behaviour of competitors. So, there is a difference between a strategy and a single move  a strategy is a kind of knowledge about a whole situation for making the best decision in terms of expected utility. It could be the main factor in winning a market but may not always bring success. Here are some of my thoughts about strategies.
Follow the leader
Follow the leader and wait for a mistake. Sometimes, to achieve success or dominate a market you need to be more innovative than your competitors. Take Apple and IBM as an example: both produce computers but IBM had a dominant market position before Steve Jobs created Apple. IBM became a monopoly through the standardization of PC computers; if Apple wanted a piece of the pie, it need to be more innovative than IBM was. Sometimes we are too focussed on what’s around us, and miss out on vital opportunities as a consequence of this failure.
Think big, but start small
A strategy for making a product. A client has started thinking about a product and has a grand view of what it should look like, but it is too spread for an MVP and is like a cloud, in that only its shape is known. To me, a product should be a snowball which, once it starts rolling, gets bigger and bigger. Be iterative; allow your products to increase with time and with you. Starting from smaller things allows you to be more sensitive to changes and to your competitors.
Rome wasn't built in a day  be patient. Sometimes a product needs time to become more mature and ready for sale. Be ready for failure: the sooner it happens, the less money it costs and the sooner you get feedback the better.
Summary
Above I presented basic thoughts about Game Theory and Mathematics. I’ve found that it makes it easier for me to think about, approach, and solve certain kinds of problems. It informs basic intuitions which allow us to make approximations of some problems. In daily situations, it can be very useful for making decisions and thinking about the consequences of acts. It’s worth mentioning that Game Theory allows the design of intelligent agents which are able to behave in a more rational way. When making your strategy, don’t forget about unpredictable events  it’s inevitable that they happen! Then you change your current strategy in order to not lose a market. Strategy is a continuous process which requires changes based on information or, perhaps, big data.