Two wrongs can make a right.
COVID-19 AND PARRONDO’S PARADOX
This one is for the losers. What if I told you that there is a way to win, even after losing? Parrondo’s Paradox states that if there exists a pair of games, each of which has a greater probability of losing than winning, it is possible to devise a winning strategy by playing those games alternately. In other words, you can combine two losing strategies to make a winning one! This counterintuitive argument was proposed by Juan Parrondo in 1996. Since then, it has been used extensively in game theory.
Let us take the simplest example. Consider two games: Game 1 and Game 2.
- Game 1: you lose $1 every time you play.
- Game 2: if the amount of money left with you is an even number, you win $3, else you lose $5.
Let’s say you start with $50. If you decide to play Game 1 only, you will end up exhausting your money in 50 tries. If you decide to play Game 2 only, you will end up exhausting your money yet again.
This is where Parrondo’s Paradox comes into the picture. If you play the games alternately (starting with Game 2), you will have a net profit of $2 for every 2 tries. (Since you start with $50, which is even, you gain $3, and you now have $53. Proceed to Game 1, lose a dollar, and your total adds up to $52, which is even. Now proceed to Game 2 again, and so forth.)
Original Parrondo’s Game:
Consider two simple coin-tossing games A & B. The player’s capital is linked with these games. If he wins, the capital increase by 1, else it decreases by 1. The games are as follows:
- Game A: played with a biased coin, with odds of losing being slightly greater than those of winning. The probability of winning is p=1/2-e, and that of losing is 1/2+e, where e is a biasing parameter with a minimal value. Consider e = 0.005
- Game B: two cases arise here:
- The player’s capital is not a multiple of 3: player tosses a biased coin, with the odds of winning being slightly greater than losing. The probability of winning is p1=3/4-e, and that of losing is 1/4+e. Let us refer to this coin as the “good coin.”
- The player’s capital is a multiple of 3: player tosses a heavily biased coin, with the odds of losing being much greater than winning. The probability of winning is p2=1/10-e, and that of losing is 9/10+e. Let us refer to this coin as the “bad coin.”
Clearly, games A and B are losing games. Hence, if a player plays only A or only B, he will lose capital. However, if he plays these two games in a specific pattern, like AABBAA…, or even in a random fashion, then it is observed that his capital increases as he plays more turns.
The result is mind-boggling: how did two losing games combine to give profit? Game A is always a losing game. But game B is played with a bad coin and a good coin. The number of times the bad coin is played can be controlled through the capital, and this number gets reduced when game A is played along with it. The effect of playing game A is to reorganize the capital in such a way that the capital is less likely to be a multiple of 3. So, the key is to control the capital by playing a combination of the two games.
Relieving Cost of Epidemic by Parrondo’s Paradox:
Because of the worldwide outbreak of COVID-19, the world has come to a standstill. As the disease spreads rapidly, so do paranoia and anxiety among the people. The only way to curb the transmission of this disease would be to impose a complete lockdown. However, it will paralyze economic activity worldwide as factories shut, bringing manufacturing to a halt, and individual activities are hampered. This will have a significant impact on the economy and individual purchasing power. But consider the case where a complete lockdown is not imposed. The disease will transmit from one person to another, increasing the number of affected individuals, and naturally, the number of casualties will also increase. This will strain healthcare services (we’ve all seen the undying efforts put in by frontline workers to save lives). Moreover, it will also affect the mental health of the people. If there is a looming threat of catching a (potentially) deadly disease every time you step out of your house, even for the bare necessities, it would naturally take a toll on you.
So, we have two conflicting scenarios: a complete lockdown, which would curb the epidemic and affect the economy, and “letting nature take its course,” which will claim lives and strain healthcare services, but the economy would thrive. Let’s analyze this situation by applying Parrondo’s paradox.
The two losing games are an open community (Strategy A), and complete lockdown (Strategy B). Also, consider a “cost” function F(t), which takes into account the loss to both society and individuals in the form of hospitalization cost, personal opportunity cost, human capital investment (defined as the economic value of a worker’s experience and skills), and the cost of risky behavior. Meanwhile, the cumulative cost ℱ(𝑡) is considered together with the “cost” per day. The cumulative cost ℱ(𝑡) is the cumulative cost from the beginning to time t. We also consider a SIADE model to understand the interaction and flow between the different population compartments during the COVID-19 pandemic.
Two different types of alternating strategies can be employed:
- Time-based switching scheme: this scheme alternates between strategies A and B based on time, i.e., the open community is implemented in the period [0, t) and lockdown in the period [t, T).
- Result-based switching scheme: this scheme alternates between strategies A and B based on the number of infections on the previous day, i.e., if the infection count is higher than a given number, say N, then lockdown will be implemented. Else, an open community will be implemented.
Cost arising from individual strategies:
Figures a-c show the effect of Lockdown, i.e., Strategy B.
Figures d-f show the effect of Open Community, i.e., Strategy A.
In figure (a), we observe that after the lockdown, D and A have reduced significantly. The first wave of epidemic strikes end around t=25, and the aggressive lockdown strategy results in the eradication of the number of people infected by the virus around t=40. Figure (b) proves that isolation results in the virus spreading less, but figure (c ) shows that the total cost starts to rise when lockdown is imposed. The cost then further increases at a faster rate beyond t=40. The initial decline is attributed to decreases in the loss of human capital and in hospitalization costs. However, this is soon outweighed by the individual opportunity cost due to isolation. Hence, the total cost F(t) will continue to increase after t = 40. This rising cost allows us to classify strategy B as a losing strategy. The same analysis applies to strategy A, leading to the same conclusion.
Cost arising from alternating strategies:
Figures a-c show the effect of the Time-Based Switching scheme.
Figures d-f show the effect of the Result-Based Switching scheme.
When alternating between the two strategies according to the time-based switching scheme, we observe that the rate of increase in population E decreases, and remains lower than if we were to employ strategy B individually. Meanwhile, the net population sizes of D and A gradually decrease in each instance of the switching. This proves that the switching strategy is useful in controlling the epidemic. We observe a downward trend for both populations A and D. Closer inspection of the population size of S and I in Figure (b) shows that time-based switching gradually increases the population in isolation I. This ratcheting effect means that no one single cost can significantly outweigh the three other cost components, and thus the cost decreases and stabilizes over time as observed in Figure ©. As the strategies are time-based, the periodic rise and fall in the number of infected individuals will be largely predictable and periodic in real life. A similar trend can be observed from the result-based switching scheme. It is worth noting that, as result-based switching is based on the number of infections from the previous day, it employs a stricter condition for switching between strategy A and strategy B. This allows populations A and D to fall significantly before a second wave is observed again, as depicted in Figure (d). This stricter control measure also results in fewer deaths.
Clearly, these alternating strategies are better than individual strategies.
Conclusion:
This article sheds light on the paradoxical result of switching between two losing strategies to lower the “cost” of an epidemic via Parrondo’s games. The alternating strategies have been applied in real life too, and are the reason we’ve avoided an extreme number of deaths and a severe setback to the economy.
The virus has been around for more than 2 years now, and people (extroverts, if I may) found it difficult to adjust to this change in lifestyle. The pandemic has been hard on everyone, and the lockdowns seem to last eternities. The days seemed to drag on and we were robbed of our abilities to recall which day of the week it was. But, like always, it will get better. Humanity always finds a way.
