<p>This is a classic principal-agent problem from game theory. The team wants to get the payoff from winning while minimising effort, while the leader wants to win while minimising the amount that they have to pay the team for the victory. Winning is a binary outcome while effort is linear, which creates the non-linearity that you refer to. The team's incentive to work harder is small if they are in the lead, or if they are far behind, because there is a small probability of victory. However, if they are just a few points behind, the incentive to exert effort is high because their  payoff goes from 0 to 1. The leader can't observe the team's effort and the team can't observe the score (information asymmetry), but the leader can lie and say they are just behind and rely on the non-linear payoffs to incentivise the team.</p>
<p>Mathematically, you can set up two maximisation problems - one for each agent - then find the optimal strategy for the principal (that is, the leader) at the intersection of the two. This strategy will be in the form of: (1) the number of points behind for the lie; (2) the payment to the team if they succeed; (3) the probability of the leader lying [thank you to @VARulle for reminding me of the possibility of a mixed Nash equilibrium]. Review principal agent problems in an introductory game theory textbook for help in setting up a suitable game.</p>

CC BY-SA 4.0
2022-08-25T09:46:14.667Z
That's not a Nash equilibrium. The team can infer that the leader has no incentive to tell the truth and will therefore optimally ignore his message.
VARulle
CC BY-SA 4.0
2022-08-25T18:27:56.607Z
@VARulle, good point. I have added a probability of lying to include the possibility of a mixed-strategy equilibrium. As long as the probability is sufficiently low, the team has the incentive to believe the leader. (This probability may in fact be a continuous distribution, or even zero depending on the team's utility function.)
finite_seed
CC BY-SA 4.0
2022-08-26T21:25:50.303Z
Hi! What exactly do you mean by payment here? Seems like all the principal does is relay information?
Giskard
CC BY-SA 4.0
2022-08-27T22:25:08.337Z
@Giskard, this is just based on the standard principal agent game, but we add in the principal's ability to lie to the team. In the formulation that I have laid out, the principal would get paid if they win, then pays some share of this to the team. The teams expected payoff is higher if the principal lies, which means that the principal doesn't need to pay them as much and can keep more of the return for themself,thus creating the incentive for the lie. You could also formulate it with set returns for both parties.
finite_seed
CC BY-SA 4.0
2022-08-28T05:28:26.237Z
"*This is a classic principal-agent problem from game theory.*" and "*this is just based on the standard principal agent game*" are in strong conflict here! Your insight about the non-linearity of payoff in effort is good, but there is almost no other shared element with the principal-agent model - the principal is not a strategic actor, but has preset random behavior (treatments), so there is no conflict of interest. There is also an aspect of "war chest" competition here that is missing in principal-agent models.
Giskard
CC BY-SA 4.0
2022-08-29T02:14:10.693Z
@Giskard, you really haven't done enough game theory. Literally every point you made is erroneous. I'm happy to address them, but will only do so if OP asks me about any of your specific points
finite_seed
CC BY-SA 4.0
2022-08-29T05:16:40.383Z
[Lol](https://economics.stackexchange.com/users/1601/giskard)
Giskard
CC BY-SA 4.0
2022-08-24T18:36:25.987Z
2022-08-25T18:24:07.700Z
2022-08-25T18:24:07.700Z
finite_seed
<blockquote>
<p>How do economists explain: &quot;If You Want to Win, Tell Your Team It’s Losing (a Little).&quot; ?</p>
</blockquote>
<p>I read <a href="https://hbr.org/2011/10/if-you-want-to-win-tell-your-team-its-losing-a-little" rel="nofollow noreferrer">this article</a>. And I learnt that—<br />
&quot;<strong>The relationship between the score and the likelihood of winning was fairly linear</strong>. For every two points a team was ahead, its chances of winning increased by about 7%—except for this major discontinuity right in the middle. Teams that were <strong>down by one point</strong> at halftime were <strong>more likely to win than teams that were ahead by one point</strong> at halftime. They won as much as 8% more often than they would have if the relationship had stayed linear.&quot;<br />
<em>A question more</em>. How can I get a mathematical explanation for the above scenario (stress on <em><strong>bolds</strong></em>) along with self-efficacy ?</p>

CC BY-SA 4.0
2022-08-24T12:53:12.400Z
Hi! What do you mean by "*a mathematical explanation*"?
Giskard
CC BY-SA 4.0
2022-08-24T12:55:03.607Z
"*How do economists explain the theory of...*" and what makes you think economists have an (economics related) explanation for this? E.g., I am not sure if the theory of relativity or the theory of inflamation causing Alzhemier's have economics related explanations.
Giskard
<mathematical-economics><utility><competitive-equilibrium><mechanism-design><fairness>
How do economists explain "If You Want to Win, Tell Your Team It’s Losing (a Little)." along with self-efficacy
Robbinspire
# How do economists explain "If You Want to Win, Tell Your Team It’s Losing (a Little)." along with self-efficacy

(Asked by: username_0 on 2022-08-24)

> How do economists explain: "If You Want to Win, Tell Your Team It’s Losing (a Little)." ?

I read [this article](https://hbr.org/2011/10/if-you-want-to-win-tell-your-team-its-losing-a-little). And I learnt that—  
"**The relationship between the score and the likelihood of winning was fairly linear**. For every two points a team was ahead, its chances of winning increased by about 7%—except for this major discontinuity right in the middle. Teams that were **down by one point** at halftime were **more likely to win than teams that were ahead by one point** at halftime. They won as much as 8% more often than they would have if the relationship had stayed linear."  
*A question more*. How can I get a mathematical explanation for the above scenario (stress on ***bolds***) along with self-efficacy ?

**Tags:** mathematical-economics, utility, competitive-equilibrium, mechanism-design, fairness
> **Comments:**
> **username_1**: Hi! What do you mean by "*a mathematical explanation*"?  
> **username_1**: "*How do economists explain the theory of...*" and what makes you think economists have an (economics related) explanation for this? E.g., I am not sure if the theory of relativity or the theory of inflamation causing Alzhemier's have economics related explanations.  

---

## 1 answers

### Answer by username_2 on 2022-08-24. Score: 2 | Accepted answer

This is a classic principal-agent problem from game theory. The team wants to get the payoff from winning while minimising effort, while the leader wants to win while minimising the amount that they have to pay the team for the victory. Winning is a binary outcome while effort is linear, which creates the non-linearity that you refer to. The team's incentive to work harder is small if they are in the lead, or if they are far behind, because there is a small probability of victory. However, if they are just a few points behind, the incentive to exert effort is high because their payoff goes from 0 to 1. The leader can't observe the team's effort and the team can't observe the score (information asymmetry), but the leader can lie and say they are just behind and rely on the non-linear payoffs to incentivise the team.

Mathematically, you can set up two maximisation problems - one for each agent - then find the optimal strategy for the principal (that is, the leader) at the intersection of the two. This strategy will be in the form of: (1) the number of points behind for the lie; (2) the payment to the team if they succeed; (3) the probability of the leader lying [thank you to @VARulle for reminding me of the possibility of a mixed Nash equilibrium]. Review principal agent problems in an introductory game theory textbook for help in setting up a suitable game.

> **Comments:**
> **username_3**: That's not a Nash equilibrium. The team can infer that the leader has no incentive to tell the truth and will therefore optimally ignore his message.  
> **username_2**: @VARulle, good point. I have added a probability of lying to include the possibility of a mixed-strategy equilibrium. As long as the probability is sufficiently low, the team has the incentive to believe the leader. (This probability may in fact be a continuous distribution, or even zero depending on the team's utility function.)  
> **username_1**: Hi! What exactly do you mean by payment here? Seems like all the principal does is relay information?  
> **username_2**: @Giskard, this is just based on the standard principal agent game, but we add in the principal's ability to lie to the team. In the formulation that I have laid out, the principal would get paid if they win, then pays some share of this to the team. The teams expected payoff is higher if the principal lies, which means that the principal doesn't need to pay them as much and can keep more of the return for themself,thus creating the incentive for the lie. You could also formulate it with set returns for both parties.  
> **username_1**: "*This is a classic principal-agent problem from game theory.*" and "*this is just based on the standard principal agent game*" are in strong conflict here! Your insight about the non-linearity of payoff in effort is good, but there is almost no other shared element with the principal-agent model - the principal is not a strategic actor, but has preset random behavior (treatments), so there is no conflict of interest. There is also an aspect of "war chest" competition here that is missing in principal-agent models.  
> **username_2**: @Giskard, you really haven't done enough game theory. Literally every point you made is erroneous. I'm happy to address them, but will only do so if OP asks me about any of your specific points  
> **username_1**: [Lol](https://economics.stackexchange.com/users/1601/giskard)  

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