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When You Just Can’t Decide on a Single Action

In Game Theory, the players typically have to make assumptions about the other players’ actions. What will the other player do? Will he use rock, paper or scissors? You never know, but in some cases, you might have an idea of the probability of some actions being higher than others. Adding such a notion of probability or randomness opens up a new chapter in game theory that lets us analyse more complicated scenarios.  This article is the third in a four-chapter series on the fundamentals of game theory. If you haven’t checked out the first two chapters yet, I’d encourage you to do that to become familiar with the basic terms and concepts used in the following. If you feel ready, let’s go ahead! Mixed Strategies To the best of my knowledge, soccer is all about hitting the goal, although that happens very infrequently. Photo by Zainu Color on Unsplash So far we have always considered games where each player chooses exactly one action. Now we will extend our games by allowing each player to select different actions with given probabilities, which we call a mixed strategy. If you play rock-paper-scissors, you do not know which action your opponent takes, but you might guess that they select each action with a probability of 33%, and if you play 99 games of rock-paper-scissors, you might indeed find your opponent to choose each action roughly 33 times. With this example, you directly see the main reasons why we want to introduce probability. First, it allows us to describe games that are played multiple times, and second, it enables us to consider a notion of the (assumed) likelihood of a player’s actions.  Let me demonstrate the later point in more detail. We come back to the soccer game we saw in chapter 2, where the keeper decides on a corner to jump into and the other player decides on a corner to aim for. A game matrix for a penalty shooting. If you are the keeper, you win (reward of 1) if you choose the same corner as the opponent and you lose (reward of -1) if you choose the other one. For your opponent, it is the other way round: They win, if you select different corners. This game only makes sense, if both the keeper and the opponent select a corner randomly. To be precise, if one player knows that the other always selects the same corner, they know exactly what to do to win. So, the key to success in this game is to choose the corner by some random mechanism. The main question now is, what probability should the keeper and the opponent assign to both corners? Would it be a good strategy to choose the right corner with a probability of 80%? Probably not.  To find the best strategy, we need to find the Nash equilibrium, because that is the state where no player can get any better by changing their behaviour. In the case of mixed strategies, such a Nash Equilibrium is described by a probability distribution over the actions, where no player wants to increase or decrease any probability anymore. In other words, it is optimal (because if it were not optimal, one player would like to change). We can find this optimal probability distribution if we consider the expected reward. As you might guess, the expected reward is composed of the reward (also called utility) the players get (which is given in the matrix above) times the likelihood of that reward. Let’s say the shooter chooses the left corner with probability p and the right corner with probability 1-p. What reward can the keeper expect? Well, if they choose the left corner, they can expect a reward of p*1 + (1-p)*(-1). Do you see how this is derived from the game matrix? If the keeper chooses the left corner, there is a probability of p, that the shooter chooses the same corner, which is good for the keeper (reward of 1). But with a chance of (1-p), the shooter chooses the other corner and the keeper loses (reward of -1). In a likewise fashion, if the keeper chooses the right corner, he can expect a reward of (1-p)*1 + p*(-1). Consequently, if the keeper chooses the left corner with probability q and the right corner with probability (1-q), the overall expected reward for the keeper is q times the expected reward for the left corner plus (1-q) times the reward for the right corner.  Now let’s take the perspective of the shooter. He wants the keeper to be indecisive between the corners. In other words, he wants the keeper to see no advantage in any corner so he chooses randomly. Mathematically that means that the expected rewards for both corners should be equal, i.e. which can be solved to p=0.5. So the optimal strategy for the shooter to keep the keeper indecisive is to choose the right corner with a Probability of p=0.5 and hence choose the left corner with an equal probability of p=0.5.  But now imagine a shooter who is well known for his tendency to choose the right corner. You might not expect a 50/50 probability for each corner, but you assume he will choose the right corner with a probability of 70%. If the keeper stays with their 50/50 split for choosing a corner, their expected reward is 0.5 times the expected reward for the left corner plus 0.5 times the expected reward for the right corner: That does not sound too bad, but there is a better option still. If the keeper always chooses the right corner (i.e., q=1), they get a reward of 0.4, which is better than 0. In this case, there is a clear best answer for the keeper which is to favour the corner the shooter prefers. That, however, would lower the shooter’s reward. If the keeper always chooses the right corner, the shooter would get a reward of -1 with a probability of 70% (because the shooter themself chooses the right corner with a probability of 70%) and a reward of 1 in the remaining 30% of cases, which yields an expected reward of 0.7*(-1) + 0.3*1 = -0.4. That is worse than the reward of 0 they got when they chose 50/50. Do you remember that a Nash equilibrium is a state, where no player has any reason to change his action unless any other player does? This scenario is not a Nash equilibrium, because the shooter has an incentive to change his action more towards a 50/50 split, even if the keeper does not change his strategy. This 50/50 split, however, is a Nash equilibrium, because in that scenario neither the shooter nor the keeper gains anything from changing their probability of choosing the one or the other corner.  Fighting birds Food can be a reason for birds to fight each other. Photo byViktor Keri on Unsplash From the previous example we saw, that a player’s assumptions about the other player’s actions influence the first player’s action selection as well. If a player wants to behave rationally (and this is what we always expect in game theory), they would choose actions such that they maximize their expected reward given the other players’ mixed action strategies. In the soccer scenario it is quite simple to more often jump into a corner, if you assume that the opponent will choose that corner more often, so let us continue with a more complicated example, that takes us outside into nature.  As we walk across the forest, we notice some interesting behaviour in wild animals. Say two birds come to a place where there is something to eat. If you were a bird, what would you do? Would you share the food with the other bird, which means less food for both of you? Or would you fight? If you threaten your opponent, they might give in and you have all the food for yourself. But if the other bird is as aggressive as you, you end up in a real fight and you hurt each other. Then again you might have preferred to give in in the first place and just leave without a fight. As you see, the outcome of your action depends on the other bird. Preparing to fight can be very rewarding if the opponent gives in, but very costly if the other bird is willing to fight as well. In matrix notation, this game looks like this: A matrix for a game that is someties called hawk vs. dove. The question is, what would be the rational behaviour for a given distribution of birds who fight or give in? If you are in a very dangerous environment, where most birds are known to be aggressive fighters, you might prefer giving in to not get hurt. But if you assume that most other birds are cowards, you might see a potential benefit in preparing for a fight to scare the others away. By calculating the expected reward, we can figure out the exact proportions of birds fighting and birds giving in, which forms an equilibrium. Say the probability to fight is denoted p for bird 1 and q for bird 2, then the probability for giving in is 1-p for bird 1 and 1-q for bird 2. In a Nash equilibrium, no player wants to change their strategies unless any other payer does. Formally that means, that both options need to yield the same expected reward. So, for bird 2 fighting with a probability of q needs to be as good as giving in with a probability of (1-q). This leads us to the following formula we can solve for q: For bird 2 it would be optimal to fight with a probability of 1/3 and give in with a probability of 2/3, and the same holds for bird 1 because of the symmetry of the game. In a big population of birds, that would mean that a third of the birds are fighters, who usually seek the fight and the other two-thirds prefer giving in. As this is an equilibrium, these ratios will stay stable over time. If it were to happen that more birds became cowards who always give in, fighting would become more rewarding, as the chance of winning increased. Then, however, more birds would choose to fight and the number of cowardly birds decreases and the stable equilibrium is reached again.  Report a crime There is nothing to see here. Move on and learn more about game theory. Photo by JOSHUA COLEMAN on Unsplash Now that we have understood that we can find optimal Nash equilibria by comparing the expected rewards for the different options, we will use this strategy on a more sophisticated example to unleash the power game theory analyses can have for realistic complex scenarios.  Say a crime happened in the middle of the city centre and there are multiple witnesses to it. The question is, who calls the police now? As there are many people around, everybody might expect others to call the police and hence refrain from doing it themself. We can model this scenario as a game again. Let’s say we have n players and everybody has two options, namely calling the police or not calling it. And what is the reward? For the reward, we distinguish three cases. If nobody calls the police, the reward is zero, because then the crime is not reported. If you call the police, you have some costs (e.g. the time you have to spend to wait and tell the police what happened), but the crime is reported which helps keep your city safe. If somebody else reports the crime, the city would still be kept safe, but you didn’t have the costs of calling the police yourself. Formally, we can write this down as follows: v is the reward of keeping the city safe, which you get either if somebody else calls the police (first row) or if you call the police yourself (second row). However, in the second case, your reward is diminished a little by the costs c you have to take. However, let us assume that c is smaller than v, which means, that the costs of calling the police never exceed the reward you get from keeping your city safe. In the last case, where nobody calls the police, your reward is zero. This game looks a little different from the previous ones we had, mainly because we didn’t display it as a matrix. In fact, it is more complicated. We didn’t specify the exact number of players (we just called it n), and we also didn’t specify the rewards explicitly but just introduced some values v and c. However, this helps us model a quite complicated real situation as a game and will allow us to answer more interesting questions: First, what happens if more people witness the crime? Will it become more likely that somebody will report the crime? Second, how do the costs c influence the likelihood of the crime being reported? We can answer these questions with the game-theoretic concepts we have learned already.  As in the previous examples, we will use the Nash equilibrium’s property that in an optimal state, nobody should want to change their action. That means, for every individual calling the police should be as good as not calling it, which leads us to the following formula: On the left, you have the reward if you call the police yourself (v-c). This should be as good as a reward of v times the likelihood that anybody else calls the police. Now, the probability of anybody else calling the police is the same as 1 minus the probability that nobody else calls the police. If we denote the probability that an individual calls the police with p, the probability that a single individual does not call the police is 1-p. Consequently, the probability that two individuals don’t call the police is the product of the single probabilities, (1-p)*(1-p). For n-1 individuals (all individuals except you), this gives us the term 1-p to the power of n-1 in the last row. We can solve this equation and finally arrive at: This last row gives you the probability of a single individual calling the police. What happens, if there are more witnesses to the crime? If n gets larger, the exponent becomes smaller (1/n goes towards 0), which finally leads to: Given that x to the power of 0 is always 1, p becomes zero. In other words, the more witnesses are around (higher n), the less likely it becomes that you call the police, and for an infinite amount of other witnesses, the probability drops to zero. This sounds reasonable. The more other people around, the more likely you are to expect that anybody else will call the police and the smaller you see your responsibility. Naturally, all other individuals will have the same chain of thought. But that also sounds a little tragic, doesn’t it? Does this mean that nobody will call the police if there are many witnesses?  Well, not necessarily. We just saw that the probability of a single person calling the police declines with higher n, but there are still more people around. Maybe the sheer number of people around counteracts this diminishing probability. A hundred people with a small probability of calling the police each might still be worth more than a few people with moderate individual probabilities. Let us now take a look at the probability that anybody calls the police. The probability that anybody calls the police is equal to 1 minus the probability that nobody calls the police. Like in the example before, the probability of nobody calling the police is described by 1-p to the power of n. We then use an equation we derived previously (see formulas above) to replace (1-p)^(n-1) with c/v.  When we look at the last line of our calculations, what happens for big n now? We already know that p drops to zero, leaving us with a probability of 1-c/v. This is the likelihood that anybody will call the police if there are many people around (note that this is different from the probability that a single individual calls the police). We see that this likelihood heavily depends on the ratio of c and v. The smaller c, the more likely it is that anybody calls the police. If c is (close to) zero, it is almost certain that the police will be called, but if c is almost as big as v (that is, the costs of calling the police eat up the reward of reporting the crime), it becomes unlikely that anybody calls the police. This gives us a lever to influence the probability of reporting crimes. Calling the police and reporting a crime should be as effortless and low-threshold as possible. Summary We have learned a lot about probabilities and choosing actions randomly today. Photo by Robert Stump on Unsplash In this chapter on our journey through the realms of game theory, we have introduced so-called mixed strategies, which allowed us to describe games by the probabilities with which different actions are taken. We can summarize our key findings as follows:  A mixed strategy is described by a probability distribution over the different actions. In a Nash equilibrium, the expected reward for all actions a player can take must be equal. In mixed strategies, a Nash equilibrium means that no player wants to change the probabilities of their actions We can find out the probabilities of different actions in a Nash equilibrium by setting the expected rewards of two (or more) options equal. Game-theoretic concepts allow us to analyze scenarios with an infinite amount of players. Such analyses can also tell us how the exact shaping of the reward can influence the probabilities in a Nash equilibrium. This can be used to inspire decisions in the real world, as we saw in the crime reporting example. We are almost through with our series on the fundamentals of game theory. In the next and final chapter, we will introduce the idea of taking turns in games. Stay tuned! References The topics introduced here are typically covered in standard textbooks on game theory. I mainly used this one, which is written in German though: Bartholomae, F., & Wiens, M. (2016). Spieltheorie. Ein anwendungsorientiertes Lehrbuch. Wiesbaden: Springer Fachmedien Wiesbaden. An alternative in English language could be this one: Espinola-Arredondo, A., & Muñoz-Garcia, F. (2023). Game Theory: An Introduction with Step-by-step Examples. Springer Nature. Game theory is a rather young field of research, with the first main textbook being this one: Von Neumann, J., & Morgenstern, O. (1944). Theory of games and economic behavior. Like this article? Follow me to be notified of my future posts.

In Game Theory, the players typically have to make assumptions about the other players’ actions. What will the other player do? Will he use rock, paper or scissors? You never know, but in some cases, you might have an idea of the probability of some actions being higher than others. Adding such a notion of probability or randomness opens up a new chapter in game theory that lets us analyse more complicated scenarios. 

This article is the third in a four-chapter series on the fundamentals of game theory. If you haven’t checked out the first two chapters yet, I’d encourage you to do that to become familiar with the basic terms and concepts used in the following. If you feel ready, let’s go ahead!

Mixed Strategies

To the best of my knowledge, soccer is all about hitting the goal, although that happens very infrequently. Photo by Zainu Color on Unsplash

So far we have always considered games where each player chooses exactly one action. Now we will extend our games by allowing each player to select different actions with given probabilities, which we call a mixed strategy. If you play rock-paper-scissors, you do not know which action your opponent takes, but you might guess that they select each action with a probability of 33%, and if you play 99 games of rock-paper-scissors, you might indeed find your opponent to choose each action roughly 33 times. With this example, you directly see the main reasons why we want to introduce probability. First, it allows us to describe games that are played multiple times, and second, it enables us to consider a notion of the (assumed) likelihood of a player’s actions. 

Let me demonstrate the later point in more detail. We come back to the soccer game we saw in chapter 2, where the keeper decides on a corner to jump into and the other player decides on a corner to aim for.

A game matrix for a penalty shooting.

If you are the keeper, you win (reward of 1) if you choose the same corner as the opponent and you lose (reward of -1) if you choose the other one. For your opponent, it is the other way round: They win, if you select different corners. This game only makes sense, if both the keeper and the opponent select a corner randomly. To be precise, if one player knows that the other always selects the same corner, they know exactly what to do to win. So, the key to success in this game is to choose the corner by some random mechanism. The main question now is, what probability should the keeper and the opponent assign to both corners? Would it be a good strategy to choose the right corner with a probability of 80%? Probably not. 

To find the best strategy, we need to find the Nash equilibrium, because that is the state where no player can get any better by changing their behaviour. In the case of mixed strategies, such a Nash Equilibrium is described by a probability distribution over the actions, where no player wants to increase or decrease any probability anymore. In other words, it is optimal (because if it were not optimal, one player would like to change). We can find this optimal probability distribution if we consider the expected reward. As you might guess, the expected reward is composed of the reward (also called utility) the players get (which is given in the matrix above) times the likelihood of that reward. Let’s say the shooter chooses the left corner with probability p and the right corner with probability 1-p. What reward can the keeper expect? Well, if they choose the left corner, they can expect a reward of p*1 + (1-p)*(-1). Do you see how this is derived from the game matrix? If the keeper chooses the left corner, there is a probability of p, that the shooter chooses the same corner, which is good for the keeper (reward of 1). But with a chance of (1-p), the shooter chooses the other corner and the keeper loses (reward of -1). In a likewise fashion, if the keeper chooses the right corner, he can expect a reward of (1-p)*1 + p*(-1). Consequently, if the keeper chooses the left corner with probability q and the right corner with probability (1-q), the overall expected reward for the keeper is q times the expected reward for the left corner plus (1-q) times the reward for the right corner. 

Now let’s take the perspective of the shooter. He wants the keeper to be indecisive between the corners. In other words, he wants the keeper to see no advantage in any corner so he chooses randomly. Mathematically that means that the expected rewards for both corners should be equal, i.e.

which can be solved to p=0.5. So the optimal strategy for the shooter to keep the keeper indecisive is to choose the right corner with a Probability of p=0.5 and hence choose the left corner with an equal probability of p=0.5. 

But now imagine a shooter who is well known for his tendency to choose the right corner. You might not expect a 50/50 probability for each corner, but you assume he will choose the right corner with a probability of 70%. If the keeper stays with their 50/50 split for choosing a corner, their expected reward is 0.5 times the expected reward for the left corner plus 0.5 times the expected reward for the right corner:

That does not sound too bad, but there is a better option still. If the keeper always chooses the right corner (i.e., q=1), they get a reward of 0.4, which is better than 0. In this case, there is a clear best answer for the keeper which is to favour the corner the shooter prefers. That, however, would lower the shooter’s reward. If the keeper always chooses the right corner, the shooter would get a reward of -1 with a probability of 70% (because the shooter themself chooses the right corner with a probability of 70%) and a reward of 1 in the remaining 30% of cases, which yields an expected reward of 0.7*(-1) + 0.3*1 = -0.4. That is worse than the reward of 0 they got when they chose 50/50. Do you remember that a Nash equilibrium is a state, where no player has any reason to change his action unless any other player does? This scenario is not a Nash equilibrium, because the shooter has an incentive to change his action more towards a 50/50 split, even if the keeper does not change his strategy. This 50/50 split, however, is a Nash equilibrium, because in that scenario neither the shooter nor the keeper gains anything from changing their probability of choosing the one or the other corner. 

Fighting birds

Food can be a reason for birds to fight each other. Photo byViktor Keri on Unsplash

From the previous example we saw, that a player’s assumptions about the other player’s actions influence the first player’s action selection as well. If a player wants to behave rationally (and this is what we always expect in game theory), they would choose actions such that they maximize their expected reward given the other players’ mixed action strategies. In the soccer scenario it is quite simple to more often jump into a corner, if you assume that the opponent will choose that corner more often, so let us continue with a more complicated example, that takes us outside into nature. 

As we walk across the forest, we notice some interesting behaviour in wild animals. Say two birds come to a place where there is something to eat. If you were a bird, what would you do? Would you share the food with the other bird, which means less food for both of you? Or would you fight? If you threaten your opponent, they might give in and you have all the food for yourself. But if the other bird is as aggressive as you, you end up in a real fight and you hurt each other. Then again you might have preferred to give in in the first place and just leave without a fight. As you see, the outcome of your action depends on the other bird. Preparing to fight can be very rewarding if the opponent gives in, but very costly if the other bird is willing to fight as well. In matrix notation, this game looks like this:

A matrix for a game that is someties called hawk vs. dove.

The question is, what would be the rational behaviour for a given distribution of birds who fight or give in? If you are in a very dangerous environment, where most birds are known to be aggressive fighters, you might prefer giving in to not get hurt. But if you assume that most other birds are cowards, you might see a potential benefit in preparing for a fight to scare the others away. By calculating the expected reward, we can figure out the exact proportions of birds fighting and birds giving in, which forms an equilibrium. Say the probability to fight is denoted p for bird 1 and q for bird 2, then the probability for giving in is 1-p for bird 1 and 1-q for bird 2. In a Nash equilibrium, no player wants to change their strategies unless any other payer does. Formally that means, that both options need to yield the same expected reward. So, for bird 2 fighting with a probability of q needs to be as good as giving in with a probability of (1-q). This leads us to the following formula we can solve for q:

For bird 2 it would be optimal to fight with a probability of 1/3 and give in with a probability of 2/3, and the same holds for bird 1 because of the symmetry of the game. In a big population of birds, that would mean that a third of the birds are fighters, who usually seek the fight and the other two-thirds prefer giving in. As this is an equilibrium, these ratios will stay stable over time. If it were to happen that more birds became cowards who always give in, fighting would become more rewarding, as the chance of winning increased. Then, however, more birds would choose to fight and the number of cowardly birds decreases and the stable equilibrium is reached again. 

Report a crime

There is nothing to see here. Move on and learn more about game theory. Photo by JOSHUA COLEMAN on Unsplash

Now that we have understood that we can find optimal Nash equilibria by comparing the expected rewards for the different options, we will use this strategy on a more sophisticated example to unleash the power game theory analyses can have for realistic complex scenarios. 

Say a crime happened in the middle of the city centre and there are multiple witnesses to it. The question is, who calls the police now? As there are many people around, everybody might expect others to call the police and hence refrain from doing it themself. We can model this scenario as a game again. Let’s say we have n players and everybody has two options, namely calling the police or not calling it. And what is the reward? For the reward, we distinguish three cases. If nobody calls the police, the reward is zero, because then the crime is not reported. If you call the police, you have some costs (e.g. the time you have to spend to wait and tell the police what happened), but the crime is reported which helps keep your city safe. If somebody else reports the crime, the city would still be kept safe, but you didn’t have the costs of calling the police yourself. Formally, we can write this down as follows:

v is the reward of keeping the city safe, which you get either if somebody else calls the police (first row) or if you call the police yourself (second row). However, in the second case, your reward is diminished a little by the costs c you have to take. However, let us assume that c is smaller than v, which means, that the costs of calling the police never exceed the reward you get from keeping your city safe. In the last case, where nobody calls the police, your reward is zero.

This game looks a little different from the previous ones we had, mainly because we didn’t display it as a matrix. In fact, it is more complicated. We didn’t specify the exact number of players (we just called it n), and we also didn’t specify the rewards explicitly but just introduced some values v and c. However, this helps us model a quite complicated real situation as a game and will allow us to answer more interesting questions: First, what happens if more people witness the crime? Will it become more likely that somebody will report the crime? Second, how do the costs c influence the likelihood of the crime being reported? We can answer these questions with the game-theoretic concepts we have learned already. 

As in the previous examples, we will use the Nash equilibrium’s property that in an optimal state, nobody should want to change their action. That means, for every individual calling the police should be as good as not calling it, which leads us to the following formula:

On the left, you have the reward if you call the police yourself (v-c). This should be as good as a reward of v times the likelihood that anybody else calls the police. Now, the probability of anybody else calling the police is the same as 1 minus the probability that nobody else calls the police. If we denote the probability that an individual calls the police with p, the probability that a single individual does not call the police is 1-p. Consequently, the probability that two individuals don’t call the police is the product of the single probabilities, (1-p)*(1-p). For n-1 individuals (all individuals except you), this gives us the term 1-p to the power of n-1 in the last row. We can solve this equation and finally arrive at:

This last row gives you the probability of a single individual calling the police. What happens, if there are more witnesses to the crime? If n gets larger, the exponent becomes smaller (1/n goes towards 0), which finally leads to:

Given that x to the power of 0 is always 1, p becomes zero. In other words, the more witnesses are around (higher n), the less likely it becomes that you call the police, and for an infinite amount of other witnesses, the probability drops to zero. This sounds reasonable. The more other people around, the more likely you are to expect that anybody else will call the police and the smaller you see your responsibility. Naturally, all other individuals will have the same chain of thought. But that also sounds a little tragic, doesn’t it? Does this mean that nobody will call the police if there are many witnesses? 

Well, not necessarily. We just saw that the probability of a single person calling the police declines with higher n, but there are still more people around. Maybe the sheer number of people around counteracts this diminishing probability. A hundred people with a small probability of calling the police each might still be worth more than a few people with moderate individual probabilities. Let us now take a look at the probability that anybody calls the police.

The probability that anybody calls the police is equal to 1 minus the probability that nobody calls the police. Like in the example before, the probability of nobody calling the police is described by 1-p to the power of n. We then use an equation we derived previously (see formulas above) to replace (1-p)^(n-1) with c/v. 

When we look at the last line of our calculations, what happens for big n now? We already know that p drops to zero, leaving us with a probability of 1-c/v. This is the likelihood that anybody will call the police if there are many people around (note that this is different from the probability that a single individual calls the police). We see that this likelihood heavily depends on the ratio of c and v. The smaller c, the more likely it is that anybody calls the police. If c is (close to) zero, it is almost certain that the police will be called, but if c is almost as big as v (that is, the costs of calling the police eat up the reward of reporting the crime), it becomes unlikely that anybody calls the police. This gives us a lever to influence the probability of reporting crimes. Calling the police and reporting a crime should be as effortless and low-threshold as possible.

Summary

We have learned a lot about probabilities and choosing actions randomly today. Photo by Robert Stump on Unsplash

In this chapter on our journey through the realms of game theory, we have introduced so-called mixed strategies, which allowed us to describe games by the probabilities with which different actions are taken. We can summarize our key findings as follows: 

  • A mixed strategy is described by a probability distribution over the different actions.
  • In a Nash equilibrium, the expected reward for all actions a player can take must be equal.
  • In mixed strategies, a Nash equilibrium means that no player wants to change the probabilities of their actions
  • We can find out the probabilities of different actions in a Nash equilibrium by setting the expected rewards of two (or more) options equal.
  • Game-theoretic concepts allow us to analyze scenarios with an infinite amount of players. Such analyses can also tell us how the exact shaping of the reward can influence the probabilities in a Nash equilibrium. This can be used to inspire decisions in the real world, as we saw in the crime reporting example.

We are almost through with our series on the fundamentals of game theory. In the next and final chapter, we will introduce the idea of taking turns in games. Stay tuned!

References

The topics introduced here are typically covered in standard textbooks on game theory. I mainly used this one, which is written in German though:

  • Bartholomae, F., & Wiens, M. (2016). Spieltheorie. Ein anwendungsorientiertes Lehrbuch. Wiesbaden: Springer Fachmedien Wiesbaden.

An alternative in English language could be this one:

  • Espinola-Arredondo, A., & Muñoz-Garcia, F. (2023). Game Theory: An Introduction with Step-by-step Examples. Springer Nature.

Game theory is a rather young field of research, with the first main textbook being this one:

  • Von Neumann, J., & Morgenstern, O. (1944). Theory of games and economic behavior.

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Nine Energy Service Downsizes Board

Oilfield services company Nine Energy Service Inc. has decided to reduce the size of its board of directors from eight to six members by the end of the year. In a statement, the company said the change would be beneficial to its strategic priorities going forward. Following the decision, Ernie Danner, Andy Waite, and Curtiss Harrell all resigned as directors effective February 28, 2025. The board unanimously appointed Julie Peffer and Richard Burnett as new directors, with Peffer starting service on March 1, 2025, and Burnett to follow on May 3. The company added that on February 28, the board chose current director Scott E. Schwinger to assume the role of Chairman effective March 1, succeeding Danner. Additionally, current director Darryl Willis was appointed as Chair of the Nominating, Governance, and Compensation Committee, effective March 1, taking over from Schwinger, who will remain a member of the Committee. There will be no alterations to Nine’s existing senior management team. “The Board remains focused on best positioning the company to drive value for our shareholders”, Schwinger said.  “Following strategic discussions, the board unanimously concluded that several new directors and perspectives would be beneficial to the company, as would a reduction in the size of the Board. The Board is very pleased to welcome Julie and Ricky. Together, they bring both financial and operational leadership, as well as deep experience and expertise in their respective fields, and I look forward to working with them”. Peffer, current CFO of BigBear.ai, brings financial and AI expertise and will offer a fresh perspective on AI’s industry impact alongside existing director Darryl Willis of Microsoft. Burnett, CEO of Silver Creek Exploration, adds two decades of oil and gas financial management and accounting experience, Nine said. Additionally, it is anticipated that current director Gary Thomas will resign

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Raizen Is Said to Hire JPMorgan for Argentina Energy Assets Sale

Brazil’s Raizen SA has begun to explore the sale of its oil refinery and network of gas stations in Argentina, according to people familiar with the matter. Raizen, a joint venture between oil supermajor Shell Plc and Brazilian conglomerate Cosan SA, has hired JPMorgan Chase & Co. to manage the sale, said the people, who asked not to be named discussing private matters. Press offices for Raizen and JPMorgan declined to comment.  The energy firm’s potential departure from Argentina would add to a growing list of multinational firms, including Exxon Mobil, HSBC Holdings Plc and Mercedes-Benz, that have chosen to sell operations in the country during the past year despite more investor optimism about President Javier Milei’s economic overhaul.  Brazil’s largest producer of ethanol fuel, Raizen is mulling divestments and slowing down expansions as higher borrowing costs of late in Brazil rattle its finances. Its Dock Sud oil refinery in Buenos Aires is Argentina’s oldest with a capacity of 100,000 barrels a day that only trails two facilities run by state-run oil company YPF SA. Raizen’s network of around 700 gas stations account for 18% of Argentina’s gasoline and diesel sales, second to YPF, which has more than half of the market. The fuel is branded as Shell. Raizen bought the assets for almost $1 billion in 2018 from Shell, which owned them outright, during Argentina’s last experiment with market-oriented reforms. The country then witnessed a period of big government from 2019 to 2023 before voting in libertarian Milei more than a year ago. He is on a crusade to deregulate the economy, in particular the energy and oil sectors. The divestment comes as Milei rips away controls on crude and fuel prices that were used to stem inflation. That was sometimes bad for refiners or drillers, depending on how

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Data center supply, construction surged in 2024 amid AI boom

Dive Brief: Data center supply in major “primary” markets like Northern Virginia, Atlanta and Chicago surged 34% year-over-year in 2024 to 6,922.6 MW, with a further 6,350 MW under construction at year-end, CBRE said in a Feb. 26 report. The data center vacancy rate in primary markets fell to 1.9%, driving up the average asking rates for a 250-to-500-kilowatt requirement by 2.6% year-over-year to $184.06/kW, reflecting tight supply and robust demand for AI and cloud services, CBRE said in its North America Data Center Trends H2 2024 report. Volume-based discounts for larger tenants “have been significantly reduced or eliminated” due to rising demand for large, contiguous spaces, while data center operators grapple with elevated construction and equipment costs and “persistent shortages in critical materials like generators, chillers and transformers,” CBRE said. Dive Insight: Surging demand from organizations’ use of AI is driving the record data center development, CBRE says. The demand is giving AI-related occupiers increasing influence over data center development decisions like site selection, design and operational requirements. These occupiers are “prioritizing markets with scalable power capacity and advanced connectivity solutions,” the report says.  Demand is also showing up in pricing trends.  Last year was the third consecutive year of pricing increases for 250-to-500-kW slots in primary markets, CBRE said. Following steady single-digit annual declines from 2015 to 2021, average pricing rose 14.5% in 2022, 18.6% in 2023 and 12.6% in 2024. Robust tenant demand, healthy investor appetite for alternative real estate assets and recent interest rate declines are among the factors fueling an exponential increase in data center investment activity, CBRE said. Annual sales volumes reached $6.5 billion in 2024 as average sale prices increased year-over-year, reflecting “the growing scale of data center campuses,” CBRE said. Five transactions exceeded $400 million last year. Notable capital market developments included

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Bonneville opts to join SPP’s Markets+ day-ahead market over CAISO alternative

Dive Brief: The Bonneville Power Administration plans to join the Southwest Power Pool’s Markets+ real-time and day-ahead market instead of a market being launched by the California Independent System Operator, BPA said in a draft policy released Wednesday. While the CAISO’s Extended Day-Ahead Market may offer greater financial benefits compared to Markets+, overall the SPP market is a better fit for BPA based on market design elements covering governance, resource adequacy, greenhouse gas accounting and congestion revenue, the federal power marketer said. Bonneville expects to make a final decision in May. The BPA’s draft decision sets the stage for the creation of two intertwined day-ahead markets in the West. “The idea that there’s some West-wide market ideal out there that we can get to today is just not something that is on the table,” Rachel Dibble, BPA power services vice president of bulk marketing, said at a press briefing Thursday. “Maybe someday, in future decades, there may be a point where we could merge into one market, but right now, there are many entities who support Markets+.” Dive Insight: The BPA’s decision will have a major effect on market development in the West. It sells wholesale power from federal hydroelectric dams in the Northwest, totaling about 22.4 GW. The federal power marketer also operates about 15,000 circuit miles of high-voltage transmission across the Northwest. The BPA mainly sells its power to cooperative and municipal utilities, and public power districts. In its draft decision, BPA rejected calls to wait for the West-Wide Governance Pathways Initiative to complete its effort to establish an independent governance framework for EDAM.  While a bill — SB 540 — was introduced in the California Legislature last month to implement the Pathways’ second phase, it “limits the availability of full operational administrative independence by requiring that the

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USA Won’t Hesitate on Russia and Iran Sanctions, Bessent Says

The US will not hesitate to go “all in” on sanctions on Russian energy if it helps lead to a ceasefire in the Ukraine war, Treasury Secretary Scott Bessent said Thursday. Sanctions on Russia “will be used explicitly and aggressively for immediate maximum impact” at President Donald Trump’s guidance, Bessent told an audience at the Economic Club of New York. The Trump administration is pressing Ukraine to come to the table for a ceasefire deal with Russia, and Bessent said additional sanctions on Russia could help give the US more leverage in the negotiations. Trump is ready to finalize an agreement that would give the US rights to help develop some of Ukraine’s natural resources if Ukrainian President Volodymyr Zelenskiy agrees to a tangible path for a truce and talks with Moscow, according to people familiar with the matter. Bessent criticized the Biden administration for not going harder on Russian energy sanctions for fear of driving up gas prices and asked what the point of “substantial US military and financial support over the past three years” was without matching sanctions. The US has paused military aid and some intelligence sharing with Ukraine in an effort to force the US ally to agree to negotiations with Russia over the end of the war. Bessent also said the US would ramp up sanctions on Iran, adding that the US will “shutdown” the country’s oil sector using “pre-determined benchmarks and timelines” and that “Making Iran broke again will mark the beginning of our updated sanctions policy.” The Treasury chief suggested that the US would work with “regional parties” that help Iran move its oil onto the market. One of those countries is likely to be Russia, which signaled earlier this week that it was willing to assist the US in talks with Iran on ending its nuclear

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Oil Gains on Truce Hopes but Closes Week Lower

Oil’s one-day advance wasn’t enough to rescue prices from a seventh straight weekly decline as the prospect of a temporary truce in Ukraine capped on-again, off-again tariff news that upended global markets. West Texas Intermediate futures climbed by 0.7% Friday to settle above $67 a barrel after Bloomberg reported that Russia is open to a pause to fighting in Ukraine, raising the prospect of a resumption in Moscow’s crude exports. US President Donald Trump earlier pressured the two warring nations to hasten peace talks and the White House signaled that it may relax sanctions on Russian oil if there’s progress. Crude also found support from a weakening dollar and US plans to refill its strategic oil reserve, but still was down 3.9% on the week. The Biden administration’s farewell sanctions on Russia have snarled the nation’s crude trade in recent months, with total oil and natural gas revenue last month falling almost 19% from a year earlier, Bloomberg calculations showed. Russia’s oil-related taxes are a key source of financing its war against Ukraine. A potential reintroduction of Russian barrels to the market comes amid a gloomy period for the supply outlook, as OPEC+ forges ahead with a plan to start reviving idled output in April. Meanwhile, Trump’s trade policies have fanned concerns about reduced global energy demand. “You’re seeing some volatility as people try to interpret what they think is going to happen and what it’s going to mean, but the bottom line is Russia has been able to sell its oil,” said Amy Jaffe, director of New York University’s Energy, Climate Justice and Sustainability Lab. Trump signed orders on Thursday paring back tariffs on Mexico and Canada until April 2. That timing coincides with a date when the president is expected to start detailing plans for so-called reciprocal duties

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Lenovo introduces entry-level, liquid cooled AI edge server

Lenovo has announced the ThinkEdge SE100, an entry-level AI inferencing server, designed to make edge AI affordable for enterprises as well as small and medium-sized businesses. AI systems are not normally associated with being small and compact; they’re big, decked out servers with lots of memory, GPUs, and CPUs. But the server is for inferencing, which is the less compute intensive portion of AI processing, Lenovo stated.  GPUs are considered overkill for inferencing and there are multiple startups making small PC cards with inferencing chip on them instead of the more power-hungry CPU and GPU. This design brings AI to the data rather than the other way around. Instead of sending the data to the cloud or data center to be processed, edge computing uses devices located at the data source, reducing latency and the amount of data being sent up to the cloud for processing, Lenovo stated. 

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Seven important trends in the server sphere

The pace of change around server technology is advancing considerably, driven by hyperscalers but spilling over into the on-premises world as well. There are numerous overall trends, experts say, including: AI Everything: AI mania is everywhere and without high power hardware to run it, it’s just vapor. But it’s more than just a buzzword, it is a very real and measurable trend. AI servers are notable because they are decked out with high end CPUs, GPU accelerators, and oftentimes a SmartNIC network controller.  All the major players — Nvidia, Supermicro, Google, Asus, Dell, Intel, HPE — as well as smaller vendors are offering purpose-built AI hardware, according to a recent Network World article. AI edge server growth: There is also a trend towards deploying AI edge servers. The Global Edge AI Servers Market size is expected to be worth around $26.6 Billion by 2034, from $2.7 Billion in 2024, according to a Market.US report. Considerable amounts of data are collected on the edge.  Edge servers do the job of culling the useless data and sending only the necessary data back to data centers for processing. The market is rapidly expanding as industries such as manufacturing, automotive, healthcare, and retail increasingly deploy IoT devices and require immediate data processing for decision-making and operational efficiency, according to the report. Liquid cooling gains ground: Liquid cooling is inching its way in from the fringes into the mainstream of data center infrastructure. What was once a difficult add-on is now becoming a standard feature, says Jeffrey Hewitt, vice president and analyst with Gartner. “Server providers are working on developing the internal chassis plumbing for direct-to-chip cooling with the goal of supporting the next generation of AI CPUs and GPUs that will produce high amounts of heat within their servers,” he said.  New data center structures: Not

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Data center vacancies hit historic lows despite record construction

The growth comes despite considerable headwinds facing data center operators, including higher construction costs, equipment pricing, and persistent shortages in critical materials like generators, chillers and transformers, CRBE stated. There is a considerable pricing disparity between newly built data centers and legacy facilities, reflecting the premium placed on modern, energy-efficient infrastructure. Specifically, liquid/immersion cooling is preferred over air cooling for modern server requirements, CRBE found. On the networking side of things, major telecom companies made substantial investments in fiber in the second half of 2024, reflecting the growing need for more network infrastructure and capacity to accommodate growing demand from AI and data providers. There have also been many notable deals recently: AT&T’s multi-year, $1 billion agreement with Corning to provide next-generation fiber, cable and connectivity solutions; Comcast’s proposed acquisition of Nitel; Verizon’s agreement to acquire Frontier, the largest pure-play fiber internet provider in the U.S.; and T-Mobile’s entry into the fiber internet market via partnerships with fiber-optic providers. In the quarter, Meta announced plans for a 25,000-mile undersea fiber cable that would connect the U.S. East and West coasts with global markets across the Atlantic, Indian and Pacific oceans. The project would mark the first privately owned and operated global fiber cable network. Data Center Outlook

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AI driving a 165% rise in data center power demand by 2030

Goldman Sachs Research estimates the power usage by the global data center market to be around 55 gigawatts, which breaks down as 54% for cloud computing workloads, 32% for traditional line of business workloads and 14% for AI. By 2027, that number jumps to 84 GW, with AI growing to 27% of the overall market, cloud dropping to 50%, and traditional workloads falling to 23%, Schneider stated. Goldman Sachs Research estimates that there will be around 122 GW of data center capacity online by the end of 2030, and the density of power use in data centers is likely to grow as well, from 162 kilowatts per square foot to 176 KW per square foot in 2027, thanks to AI, Schneider stated.  “Data center supply — specifically the rate at which incremental supply is built — has been constrained over the past 18 months,” Schneider wrote. These constraints have arisen from the inability of utilities to expand transmission capacity because of permitting delays, supply chain bottlenecks, and infrastructure that is both costly and time-intensive to upgrade. The result is that due to power demand from data centers, there will need to be additional utility investment, to the tune of about $720 billion of grid spending through 2030. And then they are subject to the pace of public utilities, which move much slower than hyperscalers. “These transmission projects can take several years to permit, and then several more to build, creating another potential bottleneck for data center growth if the regions are not proactive about this given the lead time,” Schneider wrote.

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Top data storage certifications to sharpen your skills

Organization: Hitachi Vantara Skills acquired: Knowledge of data center infrastructure management tasks automation using Hitachi Ops Center Automator. Price: $100 Exam duration: 60 minutes How to prepare: Knowledge of all storage-related operations from an end-user perspective, including planning, allocating, and managing storage and architecting storage layouts. Read more about Hitachi Vantara’s training and certification options here. Certifications that bundle cloud, networking and storage skills AWS Certified Solutions Architect – Professional The AWS Certified Solutions Architect – Professional certification from leading cloud provider Amazon Web Services (AWS) helps individuals showcase advanced knowledge and skills in optimizing security, cost, and performance, and automating manual processes. The certification is a means for organizations to identify and develop talent with these skills for implementing cloud initiatives, according to AWS. The ideal candidate has the ability to evaluate cloud application requirements, make architectural recommendations for deployment of applications on AWS, and provide expert guidance on architectural design across multiple applications and projects within a complex organization, AWS says. Certified individuals report increased credibility with technical colleagues and customers as a result of earning this certification, it says. Organization: Amazon Web Services Skills acquired: Helps individuals showcase skills in optimizing security, cost, and performance, and automating manual processes Price: $300 Exam duration: 180 minutes How to prepare: The recommended experience prior to taking the exam is two or more years of experience in using AWS services to design and implement cloud solutions Cisco Certified Internetwork Expert (CCIE) Data Center The Cisco CCIE Data Center certification enables individuals to demonstrate advanced skills to plan, design, deploy, operate, and optimize complex data center networks. They will gain comprehensive expertise in orchestrating data center infrastructure, focusing on seamless integration of networking, compute, and storage components. Other skills gained include building scalable, low-latency, high-performance networks that are optimized to support artificial intelligence (AI)

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Netskope expands SASE footprint, bolsters AI and automation

Netskope is expanding its global presence by adding multiple regions to its NewEdge carrier-grade infrastructure, which now includes more than 75 locations to ensure processing remains close to end users. The secure access service edge (SASE) provider also enhanced its digital experience monitoring (DEM) capabilities with AI-powered root-cause analysis and automated network diagnostics. “We are announcing continued expansion of our infrastructure and our continued focus on resilience. I’m a believer that nothing gets adopted if end users don’t have a great experience,” says Netskope CEO Sanjay Beri. “We monitor traffic, we have multiple carriers in every one of our more than 75 regions, and when traffic goes from us to that destination, the path is direct.” Netskope added regions including data centers in Calgary, Helsinki, Lisbon, and Prague as well as expanded existing NewEdge regions including data centers in Bogota, Jeddah, Osaka, and New York City. Each data center offers customers a range of SASE capabilities including cloud firewalls, secure web gateway (SWG), inline cloud access security broker (CASB), zero trust network access (ZTNA), SD-WAN, secure service edge (SSE), and threat protection. The additional locations enable Netskope to provide coverage for more than 220 countries and territories with 200 NewEdge Localization Zones, which deliver a local direct-to-net digital experience for users, the company says.

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Microsoft will invest $80B in AI data centers in fiscal 2025

And Microsoft isn’t the only one that is ramping up its investments into AI-enabled data centers. Rival cloud service providers are all investing in either upgrading or opening new data centers to capture a larger chunk of business from developers and users of large language models (LLMs).  In a report published in October 2024, Bloomberg Intelligence estimated that demand for generative AI would push Microsoft, AWS, Google, Oracle, Meta, and Apple would between them devote $200 billion to capex in 2025, up from $110 billion in 2023. Microsoft is one of the biggest spenders, followed closely by Google and AWS, Bloomberg Intelligence said. Its estimate of Microsoft’s capital spending on AI, at $62.4 billion for calendar 2025, is lower than Smith’s claim that the company will invest $80 billion in the fiscal year to June 30, 2025. Both figures, though, are way higher than Microsoft’s 2020 capital expenditure of “just” $17.6 billion. The majority of the increased spending is tied to cloud services and the expansion of AI infrastructure needed to provide compute capacity for OpenAI workloads. Separately, last October Amazon CEO Andy Jassy said his company planned total capex spend of $75 billion in 2024 and even more in 2025, with much of it going to AWS, its cloud computing division.

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John Deere unveils more autonomous farm machines to address skill labor shortage

Join our daily and weekly newsletters for the latest updates and exclusive content on industry-leading AI coverage. Learn More Self-driving tractors might be the path to self-driving cars. John Deere has revealed a new line of autonomous machines and tech across agriculture, construction and commercial landscaping. The Moline, Illinois-based John Deere has been in business for 187 years, yet it’s been a regular as a non-tech company showing off technology at the big tech trade show in Las Vegas and is back at CES 2025 with more autonomous tractors and other vehicles. This is not something we usually cover, but John Deere has a lot of data that is interesting in the big picture of tech. The message from the company is that there aren’t enough skilled farm laborers to do the work that its customers need. It’s been a challenge for most of the last two decades, said Jahmy Hindman, CTO at John Deere, in a briefing. Much of the tech will come this fall and after that. He noted that the average farmer in the U.S. is over 58 and works 12 to 18 hours a day to grow food for us. And he said the American Farm Bureau Federation estimates there are roughly 2.4 million farm jobs that need to be filled annually; and the agricultural work force continues to shrink. (This is my hint to the anti-immigration crowd). John Deere’s autonomous 9RX Tractor. Farmers can oversee it using an app. While each of these industries experiences their own set of challenges, a commonality across all is skilled labor availability. In construction, about 80% percent of contractors struggle to find skilled labor. And in commercial landscaping, 86% of landscaping business owners can’t find labor to fill open positions, he said. “They have to figure out how to do

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2025 playbook for enterprise AI success, from agents to evals

Join our daily and weekly newsletters for the latest updates and exclusive content on industry-leading AI coverage. Learn More 2025 is poised to be a pivotal year for enterprise AI. The past year has seen rapid innovation, and this year will see the same. This has made it more critical than ever to revisit your AI strategy to stay competitive and create value for your customers. From scaling AI agents to optimizing costs, here are the five critical areas enterprises should prioritize for their AI strategy this year. 1. Agents: the next generation of automation AI agents are no longer theoretical. In 2025, they’re indispensable tools for enterprises looking to streamline operations and enhance customer interactions. Unlike traditional software, agents powered by large language models (LLMs) can make nuanced decisions, navigate complex multi-step tasks, and integrate seamlessly with tools and APIs. At the start of 2024, agents were not ready for prime time, making frustrating mistakes like hallucinating URLs. They started getting better as frontier large language models themselves improved. “Let me put it this way,” said Sam Witteveen, cofounder of Red Dragon, a company that develops agents for companies, and that recently reviewed the 48 agents it built last year. “Interestingly, the ones that we built at the start of the year, a lot of those worked way better at the end of the year just because the models got better.” Witteveen shared this in the video podcast we filmed to discuss these five big trends in detail. Models are getting better and hallucinating less, and they’re also being trained to do agentic tasks. Another feature that the model providers are researching is a way to use the LLM as a judge, and as models get cheaper (something we’ll cover below), companies can use three or more models to

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OpenAI’s red teaming innovations define new essentials for security leaders in the AI era

Join our daily and weekly newsletters for the latest updates and exclusive content on industry-leading AI coverage. Learn More OpenAI has taken a more aggressive approach to red teaming than its AI competitors, demonstrating its security teams’ advanced capabilities in two areas: multi-step reinforcement and external red teaming. OpenAI recently released two papers that set a new competitive standard for improving the quality, reliability and safety of AI models in these two techniques and more. The first paper, “OpenAI’s Approach to External Red Teaming for AI Models and Systems,” reports that specialized teams outside the company have proven effective in uncovering vulnerabilities that might otherwise have made it into a released model because in-house testing techniques may have missed them. In the second paper, “Diverse and Effective Red Teaming with Auto-Generated Rewards and Multi-Step Reinforcement Learning,” OpenAI introduces an automated framework that relies on iterative reinforcement learning to generate a broad spectrum of novel, wide-ranging attacks. Going all-in on red teaming pays practical, competitive dividends It’s encouraging to see competitive intensity in red teaming growing among AI companies. When Anthropic released its AI red team guidelines in June of last year, it joined AI providers including Google, Microsoft, Nvidia, OpenAI, and even the U.S.’s National Institute of Standards and Technology (NIST), which all had released red teaming frameworks. Investing heavily in red teaming yields tangible benefits for security leaders in any organization. OpenAI’s paper on external red teaming provides a detailed analysis of how the company strives to create specialized external teams that include cybersecurity and subject matter experts. The goal is to see if knowledgeable external teams can defeat models’ security perimeters and find gaps in their security, biases and controls that prompt-based testing couldn’t find. What makes OpenAI’s recent papers noteworthy is how well they define using human-in-the-middle

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