In class, we learned about the Prisoner’s dilemma as a starting topic of game theory. In this game the prisoners cannot communicate with each other and they independently choose between not confessing and confessing. The researchers Yuma Fujimotoa and Kunihiko Kaneko at the University of Tokyo take it one step further to study the iterated prisoner’s dilemma. In this version, the suspects both learn from the previous game to decide on the current game.
It was believed that both players usually will learn from previous games and this would result in no player being a victim. This research proves this wrong. According to the researchers, the result of this study is that exploitive behaviour always exists, and this exploitative behaviour is one that is stable. The exploitive behaviour is when the probability of not confessing is not the same between the suspects and this is since it is sometimes more beneficial to confess, and this exploits the other player.
basin structure and payoff matrices
The research has many ways that such exploitive behaviour exists and that it depends widely on the initial conditions of the game. One such behaviour is as follows. Assume the suspects are Alice and Bob and that Alice learns Bob’s strategy. Alice can ensure that Bob does not confess since it is a better outcome than them both confessing, and this way Alice is able to exploit. It was also seen that exploitation can go both ways between the players causing more complex outcomes.
It is a fascinating result and I believe that this will be helpful in better understanding some of the outcomes in real life since exploitation is seen common in society.
Human is a social animal. We often seek to join communities with
similar interests, values and beliefs as we do. In communities, we also constantly
adopt new information and new behaviors from others. Depends on how the network
structure, the spread of a behavior might be different across the population.
In class, we have learned two hypotheses about how network
structure affects the spread of behavior. The “small world” topology says the strength
of weak ties can spread the information or a behavior father and quicker than a
network which is highly clustered. Long ties connect people who do not have
common neighbors/friends together, so the behavior can be quickly diffuse to
other part of the network from one point. The other hypothesis we mentioned in
class is that people usually require contact with multiple initial adopters
before the behavior get cascaded down. This hypothesis predicts that high
clustered network has more unnecessary ties, so it provides social reinforcement
for adoption. Therefore, they are better in promoting the behavior across the
population. Although these 2 hypotheses are important in the scientific field,
it is very hard for scientists to implement an empirical test of these predictions
due to the lack of abilities in constructing different social network
independently in real life. In 2010, a researcher from university of Pennsylvania
conducted a study in exploring the spread of a health behavior under different
network structure.
During the study, the researcher, Centola, created an
internet-based health community and hired 1528 participants to his experiment. Each
participant created an anonymous online profile with a set of health interests.
Centola matched each participant with other participants (aka “health buddies”)
in the study to construct an online health community. However, there is some
rules apply to the artificial online health community in the experiment setting.
In order to prevent from interpersonal affect and homophily, participants are
not allowed to communicate with other “health buddies” directly, but they could
receive emails from the admin in this study to provide update on the most
recent status of their “health buddies”. Participants need to make decisions on
whether or not to adopt a new health behavior such as registering for an
Internet-based health forum based on their “health buddies” choices.
The health forum is a completely new forum created just for this particular study, so no participants have heard it before as such the study can ensure that participants could become the only source of encouragement to other participants in the same study. Participants in this study are divided into a clustered-lattice network and a random network (see the graph below Figure 2). There is a high clustering in the clustered-lattice network. To ensure that each node have the same number of neighbors in both networks, the random network is created from the clustered-lattice network according to the small world model. In other word, in the random network each node has the same number of neighbours as in clustered-lattice network, but the clustering in the random network has been reduced due to the decrement in the number of ties between neighborhoods. In both settings, each participant was randomly assigned to be a single node in the network. The immediate neighbors of each node are the “health buddies” of that node, and the only information the node has is their own “health buddies” that they were connected to.
The researchers
ran 6 independent trials and for each trial the researchers allow the process
to run for 3 weeks. A random “seed node” is selected as an initiated diffusion.
The “seed node” will send signal to its neighbors encouraging them to adopt a
health behavior. In this case, a new health behavior is essentially a health
forum. For example, when a new person in the network adopt the health forum.
The system will send a notification email to all of the friends of this person
to encourage them to do the same. In other word, if a participant has multiple “health
buddies” adopt the forum, the participant will receive multiple emails. The
results of this experiment show that behavioral diffusion is strongly affected
by network structure.
The graph below shows the relationship between the time and adoption fraction under 3 different set of data. The research team plot the result from 2 different network structure to the same graph for comparison. We use N denote the population size, and Z denote the number of health buddies each person had. The solid black circle represents the adoption fraction in small world topology network. The open triangles represent the adoption fraction in random network.
Centola, D. (2010). The Spread of Behavior in an Online Social Network Experiment. Science. doi: science.1189910
There are 3
interesting findings in this experiment. From the data collected from the
research team, the greater clustering and diameter in a network (i.e. small
world topology), the more effective it is for spreading behaviors. For example,
as we can see from the diagram, the behavior can spread to a greater area quicker
in a high clustering network (see the solid black circles) than in the random
network (see open triangles).
The second interesting discovery is the more emails an individual received during the experiment, the more likely they will be adopting the same behavior. An individual is more likely to register on the health forum after receiving second email compared to only receiving one email. The third finding is related to the user commitment once they have adopted the new behavior. The data in the research showed that there is more than 30% of participants in the population who signed up after received 2 emails made return, compared to only less than 15% of adopters in the population who signed up after the 1st email made return visit. In addition, 40% of adopters receiving 3 emails made return visits. This finding suggests that the commitment of behaviors from the adopters are greatly affected by the social encouragement and reinforcement.
In conclusion, the small world topology is more effective in spreading behaviours to a larger scale compared to a random network because on average the behavior reached to 53.77% of the nodes in the high clustered network in comparison to only 38.26% in a random network. The results could help the public sector to think about which clustered network they should be targeting first when it comes to public health intervention.
Reference
Centola, D. (2010). The Spread of Behavior in an Online Social Network Experiment. Science. doi: science.1189910
After study with Game theory, there are more expanding concepts of relative game theories that is waiting for us to dig it up. Dynamic Games with Incomplete information is one of a form of Non-cooperative Game. As we have been introduced so far in lecture, Static Games represents two players are making their choices simultaneously, or playing with a unknown sequence of actions. However, In the Dynamic Games, participants play with a sequence of action, under the condition of incomplete information, each participant in the game knows which types of the other participants and the probability of occurrence of each type, unluckily participant has no idea which type of their components belong to.
According to the condition described above, Dynamic Games with Incomplete Information demonstrates that the latter players can obtain information about the previous players by observing the behavior of the previous player, thereby making his turn with the possible outcome he gets from the previous player.
Specifically, at the beginning of the incomplete information dynamic game, a participant establishes his own preliminary judgment based on the different types of other participants and the probability distribution of their respective types. When the game begins, the participant can correct his initial judgment based on the actual actions of other participants he observes. And based on this ever-changing judgment, choose his own strategy.
New equilibrium concept: Prefect Bayesian equilibrum
-Belief: The new term defined in PBE, the belief of a player in a given information set decides which node they are playing at, represented by a probability distribution over the nodes in the information-set. The strategies and beliefs must satisfy Sequence rationality and Consistency.
Gibbons Game, Page 176. I. R. (1,3) L. M. II. II. [1-p] [p] L’ L’ R’ R’ (2,1) (0,2) (0,0) (0,1) (p,1-p) are the beliefs of player II. p is the probability that player II puts on the history that player I has played L and (1-p) is the probability that player II puts on the history that player I has played M.
As we learned in class, game theory involves the reactions of its players. One of the biggest problems we are currently facing is climate change. Even though almost all factions around the globe have realized this issue, no one is working to actively solve this problem because of the negative financial effects. This is a problem that involves a global effort, yet no players are willing to take action because it is clear a dominated strategy.
After the recent earthquake in Kutch, a district in India, the government decided to involve NGOs for rehabilitation purposes. The withdrawal of the United States from the Paris Agreement has dealt a serious blow to the recovery of the Earth. As India’s economy continues to grow, the importance of its problems grows along with it. Climate change is one such problem.
India is witnessing changes in weather patterns that might be here to stay. Rains and floods are happening in October when they should be stopping in mid-September, and there is data to show that it is not an anomaly. Even so, some are still arguing about the existence of climate change. It is clear that something needs to be done now.
A suggestion is to explore a different way to use game theory to acquire the best environmental results. Large countries, huge firms, and more generally, groups that have a large impact on climate change can be asked to adopt this approach. The idea is the create simulations to find the best policy for supporting climate change. Game theory is all about tradeoffs. If we include valuable resources in this game such as land, water, and energy, it will result in a competitive yet intriguing paradigm.
A definite positive for this strategy is that this type of cooperation might make these entities think a bit more long-term. No one wants to stay in a zero-sum game. At the end of the day, we all want a greener planet. Solving problems such as water shortage, inefficient irrigation systems will greatly reduce the effects of climate change. One of the benefits of globalization is the potential ability to tackle climate change, and I have hope that we can work together for a better future.
The fascinating thing about the concept of Game Theory is its relevance in many every day decision-making in the real world. Game Theory involves players that have strategies that may result in payoffs, whether good or bad results.
These underlying principles are prevalent in the economies of China and the U.S., as difficult decisions regarding tariffs have strong impacts in the respective economies. Currently, Trump’s decisions on the current trade deal focus on increasing U.S. exports but ignore China subsidies for specific industries. The payoff of this decision is low as China is displeased with the agreement; the U.S. should seek to emulate decisions that would benefit both sides.
Trump was only agreeing with these decisions since it was evaluated by economists that state-led industrial policy was wasteful and ineffective. However, the payoff matrix changed because of a new evaluation of China’s trade and commercial practices, which state these policies are actually effective. And as a result, the payoff for the decision demanding China to abandon polices to promote indigenous companies should be much lower than what Trump’s team have predicted.
I chose to write about the economic policies related to Trump and China because of the impacts Trump’s potential payoff matrices could have to the quality of life in the U.S. It shows that no matter how broad the concept is in the course, it can always be correlated to real world scenarios.
There are several ethical theories that philosophers have come up with to describe how people should make decisions. Specifically, I will be discussing utilitarianism. There are two kinds of utilitarianism, act and rule. I will be using the definitions that the paper in the source uses. An act-utilitarian is “one who argues that each individual act should maximize the common good,” and a rule-utilitarian is “one who argues that utilitarian principles should be applied to the rules to which we appeal when making decisions.” The paper aims to use game theory to compare and contrast the decisions made by the two kinds of utilitarians.
To further explain the differences between the two types of utilitarians, the paper uses a simple game. The author makes assumptions similar to what we have seen in class, but I will quote them here:
“1)Both players must use the same decision procedure. 2)Both players are rational decision-makers. 3)Both players are able to accurately predict the outcome of a given combination of actions. 4)Both players are aware of the first three assumptions.”
The game the author uses involves two players driving towards each other, and there is enough space for them to drive by each other. Here is the payoff matrix the author provides:
Since the payoffs are the same for both players, the numbers are not repeated in each square. Using game theory, we can see that there are two pure-strategy Nash equilibria, both drivers veer to their left or their right, so they drive by each other, avoiding a collision. The author measures this outcome in terms of happiness received by each player.
Now we assume that both players are act-utilitarians, and they both know the other is an act-utilitarian per the rules of the game. Knowing this, both Tom and Jerry know the pure-strategy Nash equilibrium, but since they cannot communicate and based on their utilitarian principles since “act-utilitarianism does not suggest right over left or vice versa”, they both have no good reason to choose veering left over veering right, so the situation becomes one where each player is playing randomly. Therefore half the time they collide and half the time they pass by each other. The author concludes, and anyone would agree, that act-utilitarianism produces sub-optimal results here.
To contrast the two types of utilitarians, we now assume both players are rule-authoritarians, and both know the other is as well per the rules of the game. Again, both players know the pure-strategy Nash equilibria, but this time, either player would come up with a universal strategy for always winning this game, then make a decision based on the strategy they come up with. For this game, either player would come up with the strategy of “always veer left” or “always veer right”. This would be fine if both come to the same strategy, but since there are multiple pure-strategy Nash equilibria, we end up with the same result as with act-utilitarianism.
The author resolves this issue by saying that as a society we have created social contracts to resolve this kind of issue, namely which side of the road to drive on. For rule-authoritarians, they are bound to this rule, and so for the rule-utilitarian situation, there is never a crash. Act-utilitarians, however, are not bound to the rules, and can therefore still crash. The author considers that for this simple game, the act-utilitarians have no reason to not abide by the rule, but says that a more complex game could easily be devised where a situation could arise where they would. The author also concedes that this may not always be the case, as his rule-set limited the possibilities. If players were allowed to communicate, for example, then the outcomes would always be optimal.
I want to highlight a situation to differentiate utilitarianism from what we have used game theory for in class. Consider the following payoff matrix for some arbitrary game:
From the perspective we have taken in class, where both players want to optimize their own result, B has a dominant strategy of always playing left, then A would play Up, resulting in a pure-strategy Nash equilibrium of Up, Left. From the perspective of utilitarianism, where the aim is to optimize a collective result, a.k.a what brings the greatest good, B would still have a dominant strategy of playing Left, but A would play down.
When applying game theory to different perspectives or rule-sets, we can reach surprising or interesting results, and exploring what other ethical theories can result in using game theory can lead to a better understanding of decision making processes.
From crops to our tables, our food would not be as prevalent if these tiny winged animals did not exist. Each bee has a responsibility to keep the hive alive, and to do so, pollinate plants and crops, which we use to sustain our food production. Bees form pollination networks to flowers and other possible pollination targets which dictates the path that bees can take to each group and complete their tasks. However as of recent, the population of bees have been steadily decreasing, so this article “Graph theoretical modelling and analysis of fragile honey bee pollination networks” attempts to analyze the relationship of disappearing bees and the destruction of these pollination networks.
In this article, researchers D. Pandiaraja et al. have attempted to construct multiple directed graph models to illustrate the various networks that bees have with pollination targets. This diagram is a basic definition of the network that is trying to constructed.
In this graph, they’ve set some definitions to define the network, such as the graph P = (V, A) with vertices V associated to flowers and A is the set of arcs. where arcs are edges that point forwards and backwards. This graph P is connected if every pair of vertices has at least one path to every other flower. Furthermore, the graph P is stable if every flower x, y such that there is a path (x, y) and (y, x) with no intermediate flower in the middle. This can be partially stable if there is either a path (x, y) or a path (y, x). The maximum set of arcs that can occur is P – V where P is completely stable or partially stable.
From these definitions, they’ve come up with some pollination networks:
Some things they’ve noticed is that P1, p4, p5 becomes compromised with the loss of a single flower. When multiple arcs are lost within a graph, the amount of possible paths dramatically decrease and bees can lose their ability to actually pollinate enough flowers to sustain their hive. There are dominating flowers who have the largest cardinality in this network that can collapse the entire pollination structure if they are removed.
Where: Pct is threat to a complete graph Ppt is threat to a partial graph, Pcs is the max cardinality for a complete graph Pps is the max cardinality of a partial graph Pa is the minimum number of arcs required to make a complete graph disconnected Pb is the minimum number of arcs required to make a partial graph disconnected
In class we’ve applied game theory to just standard games, and common understandings of it can be related primarily humans and the things we do. However, game theory can be applied in a variety of situations, and one that would typically not be considered is the application of game theory on biology and animals.
Game theory is surprisingly apparent in a lot of aspects in things like zoology and botany, from things like looking for a mate, fighting, cooperation and communication! A very interesting example of this is that there exist species that will fight to compete for a female mate, but they can actually fight on top of the female, resulting in a chance for her to be injured or die (toads). Ordinarily this wouldn’t make sense as it results in a chance for death for the female these animals are fighting to the death for, but from a game theory perspective, it makes sense. From the individual male’s point of view, what he would gain from the female would be to fertilize her eggs and spread offspring. If the female gets dies, there will be no such benefit. However, if the frog will also receive no benefit if he does not fight due to a competitor as his mate will be stolen. So the optimal choice would be to fight and expose the female to a small risk of death (Hammerstein 972-973). There are a multitude of examples of animals cooperating to improve survival and benefit themselves as it is the best choice for the given animal. For example, there exists spiders that keep frogs as pets to guard their eggs in exchange for providing them with protection and food. This is the optimal choice for the spider as even though it loses food, it leads to the best benefit for it as the spider’s eggs are protected.
In conclusion, game theory can be applied in various situations not limited to just humans. It’s application is widespread and appears throughout nature all the time with biological organisms.
We’ve talked a lot about information cascades and epidemics in class so I’m going to combine them and talk about people tweeting about the ebola outbreak from a few years back. Ebola is very deadly and tragic which got many people to fear it. However, it is extremely rare and unlikely for anyone in the Western world to get it. By examining the relationships between tweets and retweets, it becomes more evident how such news spread through people, and how the news has brought both positive and negative effects. Positive effects are that people are more aware of the situation and can thus help out by donating to the Red Cross charities to help the poor infected Africans. Negative effects are that people might live in unnecessary fear which is not good for their mental health. This report doesn’t have an actual diagram of the graph of the relationship structure of tweets and retweets since it’s studying thousands of tweets. It does give statistics of the graph though and I’ll use some of them to construct the graph myself but with descriptions.
To begin, it’s stated that 91% of the ebola retweets came from the initial message while 47.5% of those retweets is one direct retweet of the initial message (height 1 of the graph). Those 47.5% of retweets have a structural virality of 2. This shows that the spread of information is in a broadcast model instead of a viral model. This does make a lot of sense since the distribution of Twitter followers follows the power law with celebrities having lots of followers who would retweet from the celebrity. Celebrity tweets are often always trending so they’re easily seen. The broadcast model has its own advantages over the viral model. Because in a broadcast model, the average person would look at a tweet that’s been retweeted a lesser number of times. Having more and more retweets and adding commentary to the retweets is more likely to spread fake news since fake news spreads faster than real news, especially on a topic like ebola that is bound to shock people. If the original tweet was made by someone reliable like Barack Obama or UNICEF, then people in a broadcast model will get more reliable information since they’re closer to the reliable source.
In conclusion, no matter what topic especially if it’s serious, people should always see from whom they’re retweeting from because fake news can spread easily so it is better for everyone to retweet from a smaller group of reliable sources.
Let’s think about a simple case first. For example, there are four advertisers 1,
2, 3, 4
Advertiser 1,2,3,4 has a profit of v1, v2, v3, v4 per click
Advertiser 1,2,3,4 bid b1,b2,b3,b4 amount
Highest bidder gets position 1, second highest position 2, and so on
Payoff to bidder at position i is then (vi-bi+1)xi, where xi is the CTR. (CTR is the number of clicks that your ad receives divided by the number of times your ad is shown: clicks ÷ impressions = CTR. For example, if you had 5 clicks and 100 impressions, then your CTR would be 5%.)
After all the advertisers know their positions and bids, they may want to change their positions according to the known information. If the person who gets position 2 want to move up to position 1, he will need to pay a price of b1+e(e is an extremely small value). But if he wants to move down to position 3. He just needs to pay b4+e. Because people at position 4 bid b4.
In conclusion, if a bidder wants to move up to position i, he will need to pay bi+e. If a bidder wants to maintain or move down to position i, he will bid bi+1+e. ————————(*). This rule (*) can be expanded to n advertisers.
But how will those advertisers really pay? All of them want to maximize their payoff. Is there any Nash equilibrium points for them. For bidder at position i, it will be a dominant strategy for him to stay at position i if he makes at least as much profit by being in the position i than in any other positions. That is:
(Vi-bi+1)xi >= (Vi-bj)xj for j<i ——–(1)
(Vi-bi+1)xi >= (Vi-bi+1)xj for j>i ———(2)
Therefore, a set of bids(bi) that satisfies both (1) and (2) of them will be a Nash equilibrium.
I think this Nash equilibrium in auction is
very interesting because it expands the Nash equilibrium to n players(bidders).
I think this topic here is closely related to the game theory topic but also expands
our thinking.
