All posts by Vivian Li

Pure vs. Mixed Strategies

The stadium lights are blinding, and the murmuring of the crowd in the stands is amplified into a deafening roar. Yet, your senses have never been more acute. The date is January 29, 2017, and are playing to win your fifth Australian Open championship title in tennis. Millions of people are watching your every movement from across the world. You are Roger Federer. Where do you place your serves across the net?

We are often unaware of the different dimensions that everyday games are comprised of. Something seemingly as simple as a serve in tennis can be dissected into many parts, both physical and mental. In this article, we are going to explore pure and mixed strategies in game theory, using tennis as an example.

What is a pure strategy?

A pure strategy is an unconditional, defined choice that a person makes in a situation or game. For example, in the game of Rock-Paper-Scissors,if a player would choose to only play scissors for each and every independent trial, regardless of the other player’s strategy, choosing scissors would be the player’s pure strategy. The probability for choosing scissors equal to 1 and all other options (paper and rock) is chosen with the probability of 0. The set of all options (i.e. rock, paper, and scissors) available in this game is known as the strategy set.

What is a mixed strategy?

A mixed strategy is an assignment of probability to all choices in the strategy set. Using the example of Rock-Paper-Scissors, if a person’s probability of employing each pure strategy is equal, then the probability distribution of the strategy set would be 1/3 for each option, or approximately 33%. In other words, a person using a mixed strategy incorporates more than one pure strategy into a game.

The definition of a mixed strategy does not rule out the possibility for an option(s)to never be chosen (eg. pscissors= 0.5, prock = 0.5, ppaper = 0). This means that in a way, a pure strategy can also be considered a mixed strategy at its extreme, with a binary probability assignment (setting one option to 1 and all others equal to 0). For this article, we shall say that pure strategies are not mixed strategies.

In the game of tennis, each point is a zero-sum game with two players (one being the server S, and the other being the returner R). In this scenario, assume each player has two strategies (forehand F, and backhand B). Observe the following hypothetical in the payoff matrix:

The strategies FS or BS are observed for the server when the ball is served to the side of the service box closest to the returner’s forehand or backhand, respectively. For the returner, the strategies FR and BR are observed when the returner moves to the forehand or backhand side to return the serve, respectively. This gives us the payoffs when the returner receives the serve correctly (FS,FR or BS,BR), or incorrectly (FS,BR or BS,FR). The payoffs to each player for every action are given in pure strategy payoffs, as each player is only guaranteed their payoff given the opponent’s strategy is employed 100% of the time. Given these pure strategy payoffs, we can calculate the mixed strategy payoffs by figuring out the probability each strategy is chosen by each player.

So you are Roger. It is apparent to you that a pure strategy would be exploitable. If you serve to the backhand 100% of the time, it would be easy for the opponent to catch on and return from the backhand side more often than the forehand, maximizing his expected payoff. Same goes for the serve to the forehand. But how often should you mix your strategy and serve to each side to minimize your opponent’s chances of winning? Calculating these probabilities would give us our mixed strategy Nash equilibria, or the probabilities that each strategy is used which would minimize the opponent’s expected payoff. In the following article, we will look at how to find mixed strategy Nash equilibria, and how to interpret them.

The Projected Effects of Dutch Disease

In 2016, the world witnessed as Venezuela fell into a large-scale economic collapse. For some, this implosion seemingly materialized from nowhere. In revision, its causes could be traced and explained by numerous factors, including economic mismanagement from its government, rigid socialist economic policies, and a fall in oil prices. A combination of the listed factors provoked a phenomenon known as Dutch disease, which plagues the Venezuelan economy in a predicament which continues to strain the country and its citizens.

So what is Dutch disease? First coined by The Economist in 1977, the term was used to describe the effects on the Dutch economy after large deposits of natural gas were discovered in the province of Groningen in 1959. The term is most commonly applied to the discovery of natural resources such as oil and gas, but generally can refer to any situation where a large flow or investment of foreign currency into a country’s economy leads to the rapid appreciation of its currency due to its rising demand. Referring back to Groningen, the discovery increased exports in natural gas, which led to a substantial inflow of foreign currency in exchange for the guilder (the Netherlands’ currency at the time).

At first glance, this phenomenon is seemingly harmless, perhaps even beneficial, to the Dutch economy. In fact, some economists argue that Dutch disease is no disease at all, as it could be said that economies should focus on exporting commodities in which it is most efficient at producing. Furthermore, a stronger guilder makes it cheaper for the Dutch to purchase foreign goods.

However, in lagging sectors such as agriculture or manufacturing (lagging describes a sector with little and/or slow growth), appreciation of a currency makes it more expensive for foreign countries to import goods from these industries, leading in a decrease in global competitiveness. Equivalently, the large inflow of currency from exports increases the money supply domestically (assuming no intervention in monetary policy), which allows for citizens to afford more domestic goods, which inevitably leads to a rise in prices. In other words, the real exchange rate rises, once again making it harder for foreign countries to afford goods from these less dominant industries.

    \[ Real Exchange Rate = E \times \frac{P_D}{P_F} \]

where E stands for exchange rate, P_D is domestic prices, and P_F is foreign prices

So what does this mean for an economy overall? A boom in one industry damages competitiveness for non-related industries globally due to the appreciation in currency, making a country increasingly reliant on a single dominant export. For countries that export an abundant resource (examples include Nigeria and Kuwait, to name a few), issues may arise when reserves dry up, it becomes economically infeasible to export, leading to a decline in revenue. The loss in revenue is difficult to recover alternatively through exports in lagging industries that have been weakened due to a resource boom.

Venezuela’s economy came crashing down when its decade-long reliance on oil exports reared its head after OPEC decided to increase supply and production, which caused global prices to drop significantly. Its citizens, who have lived prosperously through social and welfare programs funded by revenues from their state-owned oil enterprise (the PDVSA), were greatly affected by the decision. As oil made up over 90% of Venezuela’s total exports [1], the drastic drop to approximately USD 30 per barrel at the start of 2016 from around USD 50 a year before (an even more drastic decrease when compared to appx. USD 100 per barrel from the start of 2014) [2] put a strain on revenues and production, cutting welfare for its people, as well as hindering its purchasing power of imports for  essential goods such as food.

Many lessons can be derived from these studies (which vary from case to case). But, the overarching message is the importance of diversifying an economy, and allowing for a fallback should consumers and producers tinker with the stability of markets. Definitely easier said than done.