Sunday, 23 September 2018

Is Trump vs. Xi a game of chicken, or a prisoners' dilemma?

There are several famous games that we use to teach game theory, one of which is the prisoners' dilemma (which I blogged on earlier in the week). Another is the game of chicken.

In the classic chicken game, two rivals line their cars up at opposite ends of the street. They then race directly towards each other. Each rival then has two options: (1) to swerve out of the way; or (2) to speed on. If one rival swerves and the other speeds on, the rival that sped on wins and the other driver looks foolish and loses some street cred. If both swerve, they both look a bit foolish. If both speed on, then there is a horrific accident and both may be severely injured or die. The game is presented in the payoff table below, for two drivers (Driver A and Driver B).


To find the Nash equilibrium in this game, we use the 'best response method'. To do this, we track: for each player, for each strategy, what is the best response of the other player. Where both players are selecting a best response, they are doing the best they can, given the choice of the other player (this is the definition of Nash equilibrium). In this game, the best responses are:
  1. If Driver A speeds ahead, Driver B's best response is to swerve (since a loss of face is better than dying in a fiery crash - maybe not immediately, but certainly in the long term) [we track the best responses with ticks, and not-best-responses with crosses; Note: I'm also tracking which payoffs I am comparing with numbers corresponding to the numbers in this list];
  2. If Driver A swerves, Driver B's best response is to speed ahead (since winning is better than looking a little foolish);
  3. If Driver B speeds ahead, Driver A's best response is to swerve (since, again, a loss of face is better than dying in a fiery crash);
  4. If Driver B swerves, Driver A's best response is to speed ahead (since winning is better than looking a little foolish).
Notice that there are two Nash equilibriums in this game - where one driver swerves and the other speeds ahead. Both drivers prefer the outcome where they are the one speeding ahead though, so if both try to get that outcome, we end up with both drivers dying in a fiery crash. The chicken game suggests that both players, acting in their own selfish best interest (or not considering the response of the other driver), leads to the worst possible outcome.

Which brings me to this article by Ambrose Evans-Pritchard in the Telegraph UK (gated, but there is an ungated version here):
The US and China are on a combustible escalation path that can end only when there is economic blood on the floor and the political pain threshold of one side or the other has been hit.
Both think they can withstand the longer siege. Neither can retreat easily...
China has in any case stated already that it will match each round of US tariffs with a riposte in kind.
Beijing must carry out this threat or lose face, and Trump has already vowed to escalate further when it does.
It is very hard to see how asset markets priced for perfection can ignore this deranged game of chicken for much longer. The mystery is that they have not crumbled yet.
Is this really a game of chicken? It turns out that it depends on how you define the payoffs. There are two players in the game (the U.S. and China), and two strategies (enact tariffs or hold off - the equivalents of speeding ahead or swerving). So, we can represent the game easily in a payoff table.

First, let's consider the payoffs to each country. If one country enacts tariffs and the other holds off, both countries are worse off than the status quo (they both lose some gains from trade), but the country that holds off probably loses less (their exporters are a bit worse off) [*]. We'll say that the payoff to the country enacting the tariffs is "bad", but for the other country the payoff is just "not so bad". If both countries enact tariffs then all of the bad stuff happens (exporters are a bit worse off, and there are lost gains from trade). We'll say that payoff is "very bad" for both countries. If both countries hold off, then the status quo prevails - the payoff is "OK" for both countries. The game is presented in the payoff table below.


Again solving for Nash equilibrium, the best responses are:
  1. If China enacts tariffs, the U.S.'s best response is to hold off (since "not so bad" is better than "very bad");
  2. If China holds off, the U.S.'s best response is to hold off (since "OK" is better than "bad");
  3. If the U.S. enacts tariffs, China's best response is to hold off (since "not so bad" is better than "very bad");
  4. If the U.S. holds off, China's best response is to hold off (since "OK" is better than "bad").
Notice that the best response for both countries is to hold off, regardless of what the other country does. Holding off is a dominant strategy. There is one Nash equilibrium here, which is for both countries to hold off (the status quo). It is also a dominant strategy equilibrium (because both countries have a dominant strategy). The equilibrium outcome of this game is the best outcome overall.

Clearly, that isn't the game that is playing out at the moment though, so how are things different? The current game is not a game about trade, it is a game about political posturing. The players are not the U.S. and China, but Donald Trump and Xi Jinping. They want to look strong, and not appear weak (to each other, or to their respective peoples). So, the payoffs and the players are different. The game that is actually being played looks more like the payoff table below. If both hold off, the status quo prevails (the payoff is "OK" for both. If one of them enacts tariffs and the other holds off, whichever of them enacted tariffs appears strong, and the other appears weak. If both enact tariffs, the payoff is costly to the economy.


Again solving for Nash equilibrium, the best responses are:
  1. If Xi enacts tariffs, Trump's best response is to enact tariffs (since "costly" is better than "weak");
  2. If Xi holds off, Trump's best response is to enact tariffs (since "strong" is better than "OK");
  3. If Trump enacts tariffs, Xi's best response is to enact tariffs (since "costly" is better than "weak");
  4. If Trump holds off, Xi's best response is to enact tariffs (since "strong" is better than "OK").
Notice that the best response for both leaders is to enact tariffs, regardless of what the other leader does! Enacting tariffs is a dominant strategy, and both leaders enacting tariffs is both the only Nash equilibrium and a dominant strategy equilibrium. The single equilibrium is also unambiguously worse than one of the other outcomes - this is an example of the prisoners' dilemma. Notice that it is not a chicken game.

How could this become a chicken game? If costing the economy was worse than appearing weak, then that would change things around. In that case, the best response to the other leader enacting tariffs would be to hold off. There would be two Nash equilibriums - where one leader enacts tariffs and the other holds off. However, both would prefer to be the leader enacting the tariffs rather than the one holding off.

So, whether this game is a chicken game or a prisoners' dilemma depends on how you think each leader feels about appearing weak. It seems to me that both want to avoid that at all costs. In my mind, this is a prisoners' dilemma, not a chicken game.

The repeated prisoners' dilemma can be solved for the optimal outcome (both holding off), but this requires cooperation between the two leaders. In order for this cooperation to arise, each leader must trust the other (because enacting tariffs is still a dominant strategy). If we want global trade to survive this showdown, somehow we need these leaders to develop a trusting relationship. It's a pity that Trump will not be at the APEC leaders meeting - it seems like a group hug is in order!

*****

[*] This might sound surprising. For the country imposing tariffs, tariffs lead to a deadweight loss (lost wellbeing). They make domestic sellers better off, but make domestic consumers worse off by more than the gain to domestic sellers. In contrast, the market in the country that holds off has no deadweight loss. The exporting firms in that country will be able to export a bit less, but that probably doesn't have as big of a negative impact as the tariffs do on the country that imposed them.

Saturday, 22 September 2018

Safety concerns and strawberry markets

The economic model of demand and supply is remarkably robust in terms of explaining changes in prices, and as I show in my ECONS101 class, it even works (qualitatively) when the market is not perfectly competitive. Given that my ECONS101 class has a test coming up in a week and a half, I thought it might be timely to look at an example. Let's take the recent safety scares in Australia, as reported by the New Zealand Herald:
Fruit growers across Australia are reeling from 20 reports of needles found in punnets of berries, with isolated cases of banana and apple sabotage...
Mass harvests of fruit have been dumped as prices plunge, consumer demand evaporates and products are ripped from shelves.
A police operation involving 100 officers across multiple states is now under way to hunt down those responsible...
Up to 120 growers in Queensland alone — where the scare originated — have been hit by a slump in demand and a wholesale price collapse of more than 50 per cent.
Consider the market for strawberries, as shown in the diagram below. Before the sabotage, the market was operating in equilibrium with price P0, and Q0 strawberries were being traded. Following reports of needles in strawberries, consumers have product safety concerns (who wants to buy strawberries when there's a chance of a needle strike?), so demand decreases from D0 to D1. The equilibrium price falls from P0 to P1 (a "price collapse of more than 50 per cent"), and the quantity of strawberries traded falls from Q0 to Q1.


So far, so bad. But, if you're strawberry growers, what do you do with all those strawberries that aren't being demanded by consumers? You could dump them (and some have), so maybe you find someone else willing to take them. Consider the market for strawberry jam, as shown in the second diagram below. The market was initially operating in equilibrium with price PA, and QA units of strawberry jam were being traded. Then, sabotage hits the strawberry market. The price falls. Strawberries are now much cheaper to buy (not just for consumers, but for strawberry jam producers as well). The supply curve for strawberry jam shifts down and to the right (an increase in supply), from SA to SB. The equilibrium price of strawberry jam falls from PA to PB, and the quantity of strawberry jam traded increases from QA to QB.

That last implication is testable. Check the supermarket shelves in coming months, and expect to see cheaper strawberry jam.

Friday, 21 September 2018

The prisoners' dilemma and construction tenders

After my post on the construction industry yesterday, where I suggested that clients should adopt a second-price auction to reduce risk of construction firm failures, a student noted on Facebook:
Stop undercutting the competition so that companies can actually DO the jobs they claim they can do.
In yesterday's post, I neglected to elaborate on why the construction industry can't solve the problem of firms under-pricing bids by themselves. So, in this follow-up post, let's see why, using a little bit of game theory.

Consider an industry with just two construction firms (Firm A and Firm B). [*] Both firms are bidding for a construction contract, which they know will go to the lowest bidder. The firms can choose to bid high, or bid low. Both firms are choosing their bid strategy at the same time - this is what economists refer to as a simultaneous game. The game itself is laid out in the payoff table below. If both firms price low, they each have a 50% chance of winning the contract and earning a low profit (and a 50% chance of not getting the contract and facing the loss of the resources they spent preparing their bid). If both firms price high, they each have a 50% chance of winning the contract and earning a high profit (and a 50% chance of not getting the contract and facing the loss of the resources they spent preparing their bid). If one firm prices high and the other prices low, the low-price firm wins the contract for sure and makes a low profit, and the high-price firm misses out on the contract for sure and loses the resources they spent preparing their bid. Let's further assume that winning the contract for sure at a low price is preferred over a half chance of winning the contract at a high price. [**]


To find the Nash equilibrium in this game, we use the 'best response method'. To do this, we track: for each player, for each strategy, what is the best response of the other player. Where both players are selecting a best response, they are doing the best they can, given the choice of the other player (this is the definition of Nash equilibrium). In this game, the best responses are:
  1. If Firm A bids high, Firm B's best response is to bid low (since winning the contract for sure at a low price is better than a 50% chance of winning the contract at a high price) [we track the best responses with ticks, and not-best-responses with crosses; Note: I'm also tracking which payoffs I am comparing with numbers corresponding to the numbers in this list];
  2. If Firm A bids low, Firm B's best response is to bid low (since a 50% of winning the contract at a low price is better than certainly losing the resources spent preparing the bid);
  3. If Firm B bids high, Firm A's best response is to bid low (since winning the contract for sure at a low price is better than a 50% chance of winning the contract at a high price); and
  4. If Firm B bids low, Firm A's best response is to bid low (since a 50% of winning the contract at a low price is better than certainly losing the resources spent preparing the bid).
Note that Firm A's best response is always to bid low. This is their dominant strategy. Likewise, Firm B's best response is always to bid low, which makes it their dominant strategy as well. The single Nash equilibrium occurs where both players are playing a best response (where there are two ticks), which is where both construction firms choose to bid low.

Notice that both firms would be unambiguously better if they bid high. However, both will choose to bid low, which makes them both worse off. This is a prisoners' dilemma game (it's a dilemma because, when both players act in their own best interests, both are made worse off). Both firms will choose to bid low, and whichever firm wins the contract will be at risk of having bid too low and suffering the winner's curse, as I noted yesterday. This is why the construction industry cannot solve this problem on its own.

Of course, the simple example above assumes this is a non-repeated game. A non-repeated game is played once only, after which the two players go their separate ways, never to interact again. Most games in the real world are not like that - they are repeated games. In a repeated game, the outcome may different from the equilibrium of the non-repeated game, because the players can learn to work together to obtain the best outcome.

However, cooperative strategies will not work in the construction firms' dilemma game, because such cooperation is illegal collusion. The firms would be subject to prosecution by the Commerce Commission for cartel behaviour.

So, to reiterate yesterday's conclusion, it is up to the clients of construction firms to solve this issue:
We need to ensure that sustainable contract prices are being paid, and the current system is clearly failing.
*****

[*] Limiting ourselves to two firms makes this example easy to follow, but it would work much the same if we had 20 or 200 firms (albeit being much harder to create a payoff table for!).
[**] This seems unlikely in the case of two firms, where there is a 50% chance of winning the contract, and a 50% chance of wasting time preparing the bid. However, if there are ten firms bidding, then there is a 10% chance of winning the contract, and a 90% chance of wasting time, which makes this preference seem more likely.

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Thursday, 20 September 2018

The winner's curse and construction tenders

Finally, some sanity in terms of writing about the problems the construction industry is facing. John Walton wrote in the New Zealand Herald earlier this week:
Right now, of course, the construction industry is indeed booming - not just in housing, but also in commercial and infrastructure development. But companies are continuing to fail. Why?
The answer lies in the construction process and a misunderstanding of the roles of the participants. At its simplest, the owner provides the site, resource consents, designs and pays for the work. The contractor organises the work to the design and to the required legal standards, for the agreed price within the allocated time. Price and time are adjusted for unforeseen events and for changes instructed by the owner. To this extent, construction contracts legislate for uncertainty.
That uncertainty is exacerbated by an incomplete understanding of other project risks (ground conditions and supply chain issues like subcontractor and supplier pricing and availability) and unrealistic expectations on the part of owners, particularly that they can fill in the gaps in the design and instruct changes at their whim without cost consequences. The tender process encourages this opportunistic behaviour by forcing contractors to compete on incomplete, or simply unrealistic or unfair contract terms.
Contractors try to introduce some balance by excluding risks from their bids. They must then rely on the claims process to protect their margins.
This can turn the pricing process into something of a lottery. Typically, the cheapest price wins, which all too often is submitted by the contractor with the greatest appetite for risk, coupled with the most optimistic expectations for making claims under the contract.
Following contract award, managing design development, construction and capricious owner changes to the design becomes a considerable headache for contractors who need to be able to meet construction costs, pay subcontractors and protect their already slim margins.
Every time the issue of financially troubled construction companies comes up (and it seems to be coming up a lot lately), it makes me think of the winner's curse.

Consider a group of construction firms, tendering for a construction contract. Rational construction firms who have complete information about the project and associated risks (and with the same tolerance for risk) would all have the same expectations about the costs of completing the contract, so all would bid the same. However, not all construction firms have complete information (as Walton noted above) and not all firms have the same tolerance for risk. The construction firms may make random errors in determining their costs (or margins) and risks associated with the contract, so all of the construction firms will expect different completion costs (and margins and risks) associated with the contract. For firms with similar tolerance for risk, differences in expected completion costs (and margins) arise randomly - some will overestimate the completion costs (or underestimate the risk) of the contract, and some will underestimate the completion costs (or overestimate the risk or margins). Those who expect low completion costs will bid low for the contract, and those who expect high completion costs will bid high for the contract.

The real problem arises when the contract goes to whichever construction firm bids the lowest for the contract. This will be the firm that has most underestimated the completion costs, because they will be the firm that bid the lowest. The chances are high that, if there are enough construction firms entering bids, the eventual winner will have underestimated the completion costs compared with the true costs, and hence will not receive enough to cover their true costs. This is what we refer to as the winner's curse.

The problem is that consumers of construction firms' services are too focused on looking for the lowest bid. This virtually guarantees the problems we are facing in the construction industry. Walton's piece concludes:
The industry has a choice. Either it accepts that designs and prices will change and pay contractors accordingly, or take the time to remove contract uncertainties before fixing the price and instructing work to commence. Experience here and overseas would suggest that a combination of the two works best.
Walton is not very clear on his proposed solution. So, let me offer two options.

First, when a construction contract is put up for bids, the decision could be made independent of price. That removes the incentive to bid too low. Construction firms can then be realistic about the costs and risks they face, when preparing their bids, without worrying about a high bid ruling them out. Of course, it probably creates an incentive to bid too high, precisely because a high price won't rule the firm out of the process.

Second, adopt a variant of a second-price auction. Give the contract to the lowest bidder, but pay them the amount that was asked by the second-lowest bidder. Or, if the contract is not given to the lowest bidder, still pay an amount for the contract equal to the next highest bidder's offer. This ensures that the client isn't paying well over a 'fair' price for the contract (which would likely be the case for the first option), while providing some additional space for the successful firm, which has likely underestimated the completion costs. If second-price isn't enough, a third-price auction would further limit the chance of construction firms being underpaid because of their inability to accurately estimate costs.

Something clearly needs to be done. We don't want construction firms falling over mid-contract, and in a small market like New Zealand we can't afford to have the market dominated by only a couple of players. We need to ensure that sustainable contract prices are being paid, and the current system is clearly failing.