Sequential games and backward induction

My studies on the Yale web course on game theory have progressed again. The latest topics were sequential games and, as Mr. Polak put it, one of the most important concepts one might learn at Yale, backward induction.

Sequential games are games, where one player takes action first and the second one acts afterwards, knowing what the first player did. Both players also know, that the game is sequential and that the first player’s action and its results will be known to the second player before he takes action.

The key to solving sequential games is backward induction, a concept that is quite simple and very powerful. The idea of backward induction is to picture the end of the game, the last move. Whoever makes the last move, will want to maximize their payoff, as usual. When the last move is found out this way (there are potentially many options), the move before that can be solved, when considering the payoffs of the player making the second to last move: when this player knows that the player going last will maximize his payoff, he is under a constraint when maximizing his own payoff. Similarly we can work our way backwards to the beginning of the game and anticipate the first move, too. Sequential games are often modeled as decision trees.

Although backward induction is in principle easy to use. building a decision tree may be impossible due to the infinite number of possible strategies players might play. However, in such cases it may be possible to use other tools, such as calculus in optimization problems, to find out the best responses using backward induction.

In addition to backward induction, Mr. Polak also discusses moral hazard, incentives, commitment and first-mover advantage in his lectures 13 and 14. In summary the most important lessons, apart from backward induction, are:

  1. Moral hazard can prevent players from reaching the best possible outcome, since their incentives are not aligned, e.g. a creditor and a debtor.
  2. Tweaking the game and the incentives, e.g. by reducing the payoffs of some strategies, mutually better outcomes can be achieved, as incentives become aligned, e.g. a collateral for debt reduces the risk of default.
  3. If in a sequential game you make a public, irreversible commitment to play a certain strategy, you may force your opponent to play a certain strategy, thus securing your own payoff, e.g. a company announcing to build a new plant before their competitor and consequently increasing the market capacity to 100%, and also winning market share.
  4. First-mover advantage is not always prevalent, and sometimes there is a second.mover advantage, e.g. in a sequential version of rock-paper-scissors. Sometimes one might also inadvertently make oneself into the second-mover and lose the sought first-mover advantage.

For you to remember the most important of the concepts just presented, I will repeat it one more time, backward induction. Its the answer to all questions, as Mr. Polak mentions during his lecture 13.

 

Lose your way and find nature

As a habit, I go for long walks during the weekends, usually on a Sunday. I like to take them in the woods, since the trail network here is well kept, well marked and very dense. Therefore it is very difficult to get lost and it is possible to flexibly adjust your route on the fly.

As the trail network is so good, I sometimes start my walks, or hikes, by taking a path unknown to me and start climbing uphill, since the scenery around here from almost any elevated position is quite beautiful. After some time I then take the next path downwards and start heading back home. In effect, as the trail network is so dense and well marked, I usually sooner than later end up on my way back to a road taking me straight to the right direction without any detour.

Today I started my hike on a new path, climbing upwards along the mountain side. The weather was a bit rainy and the air was moist and quite misty. I climbed higher, and among the mist and all the greenery of spring I had the feeling of being in a jungle. On multiple occasions I was expecting to see a gorilla slowly coming my way down the path second, to see first its hulking outline in the mist and finally its features getting sharper as it would get nearer.

After about an hour of climbing I took a path going downwards, and it lead me to a paradise. It was quite warm, the relative humidity can not have been much below 100%, and the fresh green of spring in the woods was very intense. The air was fresh, I saw no people and could hear and feel the rain drops hit the foliage. In the distance I heard a waterfall rumbling. Walking down the road, the sound of the waterfall got stronger, and soon I was looking down to a gorge below me, into which multiple small waterfalls were pouring the rain coming down from the sky. I walked down the road by the gorge and spotted multiple waterfalls, small rapids and beautiful rock formations. This place was a good hour away from my house door by foot, and one of the largest waterfalls even had a name sign standing before it. Yet, before today I had no idea that such wonders of nature were hiding so close to me.

Getting lost and finding something unexpected, something that you are not looking for, is a wonderful experience and shows how much more there is than we can want or even imagine. This pertains not only to nature, but to all areas of life. Sometimes, and maybe even not so seldom, it is good to take the course towards the unknown, as long as the risks are known and under control: I wouldn’t go wandering half-aimlessly in woods in completely unknown terrain and without the assurance of adequate possibilities to change course as needed. That said, without risk, without trying, there is nothing new available and many treasures will go by unnoticed. I hope you enjoy the pictures below from my hike last weekend. Unfortunately my photographing equipment does not do justice to what I saw today, so you just might have to go and find your own treasure yourself to get the real feeling.

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First day of spring

Yesterday was, at least for me, the first day of spring. It was, about 20 degrees centigrade, birds were singing and it was a great day for doing some hiking: I spent nearly four hours outdoors, wandering in the forest. The nature is waking but the flora was still waiting to really start blooming. Hopefully the next weekend will already provide some more colorful landscapes to be admired and photographed, although I am lousy and lazy at taking pictures.

I remember writing in January that I was already feeling a bit springy then, but now the days are really getting warmer after a cooler period. The snow has also mostly melted even on the elevated areas, so the spring is definitely here.

Even if I did not take any photos in the wilds during the weekend, I managed to catch today a few nice pictures of a blooming magnolia when I was getting back from work. Actually I was surprised to see it and other tress and bushes in full bloom, as I just yesterday thought that the spring is now only starting but no, it’s already advancing quickly and taking long strides.

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Nothing more to add this week, since I spent most of the weekend doing some administrative work for our sports club. The competition season is nearing, the first competitions will take place in mid-May, and I will be spending more and more time training. I still hope to be able to allocate enough time for the Yale course on Game Theory, it’s a really good course.