Games Worth Playing!

What were the games that you  played as a child? I used to play monopoly and chess! They were a lot of fun! Snakes and ladders, Ludo, Hide and Seek, chor police (thief-police) were just some of the games that Indian children grow up with. Every culture has its own games to entertain its children with. However, as we grow up we don't have too much time for these fun games anymore. So what kind of games do adults and large organizations play? The interesting thing is that as we grow up so do the games. They become rewarding to the winners and  punishing to the losers. Firms and corporations play these games all the time but they don’t play for fun, they play to win. Lets start with a linear manner of thinking.

What is a Game?

A game is any interaction between two entities (people, firms, micro-organisms) that have the power to make decisions (eat or sleep/produce or not produce/multiply or not multiply etc).

To understand every game we must ask four essential questions

• ·      Who are the players?
• ·      What are the strategies?
• ·      What is the information available to each?
• ·      What are the payoffs for each outcome?

Who are the players?

In the big world, the players are firms, households, countries, people’s groups and multi-national corporations. These players are constantly competing with each other to try and get a bigger slice of the pie. The concept of the player comes with ‘individual decision making’. If an entity can take its own decisions then it is ready to be a player in the many games that exist in the world.

What is a strategy?

A strategy is a part of the large number of decisions that an individual can take. Samsung’s choice of pricing for its Galaxy S4 is one of the many possible strategies that it could have taken. A family’s decision to go for a movie on Saturday is one of its many strategies; they could have gone out for dinner or stayed at home too. Put simply, a strategy is an option, a plan and a possible choice.

What is an outcome?

An outcome is a result. When you checkmate my king, the result is that you have won. When Neymar scores a goal in the last seconds in a football match and  helps Brazil win the match, he has changed the outcome. Outcomes emerge from conflicting or co-operating strategies. Every player makes a move, either all together or one after another, to produce a result.

What is a payoff?

Every outcome has different payoff for each player. If you have won the chess game then I will be sad and you will be happy. So the payoff for me is less than the payoff for you. If I had won (another outcome) then the payoff for me would be greater than your payoff. Similarly, when Brazil wins the match the payoff for the Brazil team is higher than the payoff for the losing team; Brazilian players are very happy!

What is the information available?

In every game, there is some amount of information available to the other players. Sometimes when games are simpler, such as Hide and Seek, all the players know each other’s payoffs and possible set of actions. In this case, there is symmetry and common knowledge. However, in some other games there is a lot of asymmetry in the amount of information available. Some countries know more about other countries. Some firms hide some valuable information from others. Every game has a distribution of information that may or may not  be equitable.

What is Game Theory?

Game Theory is a mathematical theory that developed in the 1950s and tries to find solutions and reasons for certain solutions to games. These games can be economic, biological, logical or even political. What must be remembered is that game theory uses mathematical tools and not economic, political or biological tools to solve games.

Zero Sum Games

Imagine a game between two players involving a cake. The payoff to each player depends on the amount of cake they eat. This means that if a player eats more cake, her payoff will naturally be more. Now the players divide the cake equally or unequally but no piece of the cake goes uneaten. If so, then how does one player try to improve her payoff? By getting more cake, which means that the other player must give up some cake. One’s loss is another’s gain. This is a pareto optimal because we cannot redistribute the cake seeking to improve one person’s gain without harming the other. This kind of situation is an optimal because the combined payoff to the two players cannot be improved. This optimal was named after Vilfredo Pareto who studied it.

Game 1: Cake Cutting

OUTCOME 1

Payoff to me: 300 Grams of Cake (Yummy!)

Payoff to you: 100 Grams of Cake

Total Payoff to us: 300+100=400 Grams of Cake

OUTCOME 2

Payoff to ME: 200 Grams of Cake

Payoff to YOU:200 Grams of Cake

Total Payoff : 200+200=400 Grams of Cake

OUTCOME 3

Payoff to ME: 50 Grams of Cake

Payoff to YOU: 350 Grams of Cake

Total Payoff: 400 Grams of Cake

Pareto Optimal: All the outcomes 1,2, 3 are pareto optimal since the total payoff to both of us is the same for each outcome (400 grams). This is a zero sum game because what one player loses the other gains & what one-player gains the other loses. There is a net gain of zero and a net loss of zero.

Examples of Zero Sum Games : Poker, Matching Pennies, Go, Chess, Monopoly

(Try to prove why)

Non-Zero Sum Games

Zero Sum games are a bit boring to economists. They represent Win-Lose type of games, whereas economists are looking for Win-Win type of games. This is why there is a lot of fascination with Non Zero Sum Games. These type of games have some outcomes that are better than the other outcomes. This is where Game Theory tries to find a solution. There can be many kinds of solutions depending upon who is using Game Theory. For example if a firm is using game theory, the solution it wants to find will be one that allows it to maximize profits, but if a government wants to use game theory to provide welfare for its people, the solution it is looking for is a pareto optimal that is equitable. On other occasions academicians want to just explain why games end up in the way they end up, so the solution that they are looking for involves finding an outcome that is a sort of equilibrium.

Solution 1: Dominance

A strategy is considered dominant if it is better than all other strategies available to a player for any strategy played by other players. Which actually means that if a strategy is dominant then all possible payoffs from playing that strategy is greater than the payoff from any other strategy. Here an example will explain best.

Game 2: Newspaper Wars

Imagine two newspapers competing with each other- News of the World (NOW) and The Guardian. Lets assume that there are no other players but these two. There are only two events that could help these two newspapers win the market share. Either they write a story about a recent survey about AIDS that has broken out in a nearby town, or they write about the death of a major pop singer Justin. Each player has two strategies, either write about AIDS or about Justin. However, as we can all see, the AIDS story is much more significant than the other story. So if one of them writes about Justin when the other writes about AIDS, it will lose the market share. Here are the payoffs given in a di-matrix.

The payoffs are mentioned within the boxes. There are two numbers in each box. Within each box, the number to the left is the payoff for NOW and the number to the right is the payoff for The Guardian.

There are 4 outcomes to this game.

•  Both write about the AIDS scare, and receive about 50 Million each in profit.
•  Both write about Justin, and receive about 10 Million each in profit.
•  The Guardian writes about AIDS, receives about 30 Million in profit while the NOW writes about Justin and receives about 20 Million in profit.
•  The NOW writes about AIDS, receives about 30 Million in profit, while The Guardian receive only 20 Million for writing about Justin.

What happens here is that  writing about AIDS is better than writing about Justin, whatever the other newspaper may choose to do so. For every outcome it is better for any newspaper to write about AIDS because the payoff is always higher. Hence, writing about AIDS is a dominant strategy. In this game if you notice for both NOW and The Guardian, writing about AIDS is proving to be better. Thus we have a case of a dominant equilibrium. The solution here would be that both the newspapers write about AIDS. The solution would be different if writing about AIDS was not the dominant strategy for both of them. Since, luckily writing about AIDS pays off well we will not be hearing what happened to Justin, the poor pop singer.

Solution 2: Nash Equilibrium

Sometimes in a game no strategy is dominant. If the other players play option A, maybe its best to play option B. And if the other players play option B, its best to play option B. In such scenarios there exist no dominance. So does that mean there is no solution? No! Sometimes solutions do exist without dominance, but then this is a different type of solution. Nash Equilibrium was named after John Nash, the Nobel winning mathematician who even had a movie made after him - A Beautiful Mind.

Game 3: Lover's Squat

My girlfriend and I regularly play this game. We played this game just a few months back when Iron Man 3 released and I was desperate to go see it. She on the other hand wanted to see “Ashiqui”, a romantic Hindi film. The only thing worse than seeing the movie that we did not want to do was to see any movie alone.

So in this game there are two players, my girlfriend and I. 

We have two strategies each, go see Iron Man 3 or see Ashiqui. Here are the payoffs.

As you can see we don’t like watching the movie alone. Also it should be observed that there is no dominant strategy in this game. It is better for me to go to Iron Man 3 if she goes to Iron Man 3 but it is better for me to go to Ashiqui if she is going to Ashiqui. It is the same for her.

Thus here we see a lack of dominance. Yet, a solution exists. Consider the outcome if both of us go to see Iron Man 3. In this case, none of us would want to change our strategy. This is Nash Equilibrium, since this is the best response to the other player’s response. It is mutually enforcing. There is however another Nash Equilibrium. Observe the other case wherein both of us go to see Ashiqui. In this case too, we have equilibrium. Neither of us would want to change plans.

Game Theory has offered us two solutions, either we see Iron Man 3 together or see Ashiqui together. The real world result was that we went to see Ashiqui because she made the game from a simultaneous one to a sequential game. Here she moved first and bought tickets for the movie beforehand so I had to comply. This is a move called a commitment. Where a player limits her options in order to disqualify certain Nash equilibrium.

Solution 3: Nash Equilibrium & Dominance

Game 4: War and Peace

In the 13^th century there was born a child who would be one of the world’s greatest war strategist. He would burn down cities and empires, waging war on everyone and everything that he saw. His name was Genghis Khan. He was born at a time when countries lived in peace with each other. He was born in a time when people were happy with the peace. So why did he wage war? Game Theory has an answer.

Let there be two countries who are the players, Mongolia (Genghis Khan) and the Jin Dynasty. They have two strategies, go to war or stay in peace.                                          

The numbers within the boxes represent the relative benefit to the empires. There are 4 outcomes to this game, and each outcome has two payoffs for each Genghis Khan and the Jin Dynasty. The number to the left is the payoff for the Jin Dynasty and the payoff to the right is the payoff to Genghis Khan.

• ·      Outcome 1: Both go to war, both obtain a payoff of 1
• ·      Outcome 2: Both stay in peace, both obtain a payoff of 2. And this seems like the best outcome (Pareto optimal).
• ·      Outcome 3: Genghis Khan declares war (3) and Jin Dynasty does not declare war (0)
• ·      Outcome 4: Jin Dynasty declares war (3) and Genghis Khan stays passive (1).

Outcome 2 seems to be the most beneficial to both, but as game theory tells us is that Outcome 1 is the Nash equilibrium. If one player wants to make war then the best strategy for the other player is to also make war. There is also dominance in this game. Playing War is a dominant strategy over Peace. This is one scenario where there is both mutual dominance solution and Nash equilibrium.

Here Jin Dynasty is a peaceful kingdom, usually preferring to live in peace. Genghis Khan is the aggressive one who gives into temptation and declares war. As we can see, the Nash Equilibrium is at (War,War) or (1,1) since war is a dominant strategy. Going to war is always better than being peaceful, regardless of what other player does. Hence, the Jin Dynasty became a player who was peaceful until Genghis Khan arrived. Just as how game theory predicts, the course of history was shaped by many wars of this kind.

Sequential Games

Till now we were only discussing games wherein players made their moves simultaneously. They didn't really know what the other person had played, and could only make an educated guess about it. But not all games are simultaneous. The dimension of time can make games a little more complex. Sometimes, a player plays first and then is followed by another player. Sometimes business firms act one after another instead of together or all at once. Countries make early moves that change the course of history. Games need not be static, but can be dynamic in nature.

Game 5: Price War without credible threat

 These sort of games are displayed in a game tree. At every decision the tree branches out. At the end of the branch, if no other player will play a move then the payoffs for each outcome are depicted. This is the extensive form.

Imagine a situation where Firm A has a monopoly in a market. There is a Firm B which wants to enter the market and directly compete with Firm A. This would be harmful for Firm A since it would lose market share to Firm B. Firm A has an option in this case, it can begin lowering prices and attempt to begin a price war wherein both firms will compete to lower prices to attract customers. This is a situation which both firms will choose to avoid. Firm A uses this as a threat to try and dissuade Firm B from entering. Let us see how “threats” are found to be credible or not depending on the payoffs.

There are two players: Firm A and Firm B

They have different strategies: Firm B can choose to enter or not. Firm A can choose to start a price war or maintain status quo (not start a price war).

Here are the outcomes, represented on a game tree.

Figure 1: Price War

There are three outcomes,

• ·      Outcome 1: Firm B Enters the Market, Firm A begins a price war and both firms are able to only collect 2 Million $ in profits each.
• ·      Outcome 2: Firm B Enters the Market, Firm A does not start the price war and both firms earn a profit of 5 Million $.
• ·      Outcome 3: Firm B does not enter the Market, Firm A need not begin a price war, and earns 10 Million $ while Firm B earns zero.

In case of Outcome 1, Firm A will want to change strategies and not start a price war since it will gain 5 M $ instead of 2 M $.

In case of Outcome 3, Firm B will want to change strategies since it may earn 2M$ or 5 M$ instead of the 0 M $ it is earning currently.

In this case, the Nash equilibrium would occur at outcome 2 (5 M, 5 M) since at this outcome neither Firms will want to change strategies. Hence, the threat of the price war is not credible enough and Firm B will enter the market.

Game 6: Price War with credible threat

The rules and participants of the game remain the same but the payoffs are now changed. We will observe that since the payoffs are changing, the threat of price war may become credible.

Here we observe that Firm B will go into a loss when Firm A begins a price war. We also observe that Firm A obtains larger profit from starting a price war than from not starting one, in the event that Firm B enters the market. Here Firm B realizes that if it enters the market, Firm A will always start a price war. Here the threat is credible enough and Firm B stays out of the market fearing a loss of 1 Million $.

Understanding ourselves...

Game Theory is extremely important to our lives since we are always playing a game. We sometimes play a game with other players in our lives. These other players may be rivals, friends, family or even acquaintances. We keep an eye on our own payoffs and on the other players. We use the information available at present to determine which strategies might be better. Be it picking up an umbrella for the possibility of rain or playing poker at a table, the existence of games in our life cannot be undermined. We are able to understand ourselves and societal choices through game theory. We can gauge sometimes how society and the world will react to our actions. And also, we learn about how to improve conditions of all players in a non-zero sum game.

Games are for everyone!

Economics is for everyone!

-Pranjal Rawat